Classical approximation of a linearized three waves kinetic equation
Abstract
The research of the author is supported by grants PID2020-112617GB-C21 of MINECO and IT1247-19 of the Basque Government. The hospitality of IAM of the University of Bonn, and its support through SFB 1060 are gratefully acknowledged. The author thanks Pr. M. Valle at the Universidad del País Vasco (UPV/EHU) for enlight- ening discussions. The author is also grateful to the referees for their careful reading of the manuscript, their comments and suggestions.
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Journal of Functional Analysis 282 (2022) 109390 Contents lists available at ScienceDirect Journal of Functional Analysis www.elsevier.com/locate/jfa Classical approximation of a linearized three waves kinetic equation M. Escobedo Departamento de Matemáticas, Universidad del País Vasco (UPV/EHU), Apartado 644, 48080 Bilbao, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 9 October 2020 Accepted 25 December 2021 Available online 21 January 2022 Communicated by Benjamin Schlein Keywords: Three waves collisions Classical approximation Cauchy problem Bose gas The purpose of this work is to solve the Cauchy problem for the classical approximation of an isotropic linearized three waves kinetic equation that appears in the kinetic theory of a condensed gas of bosons near the critical temperature. The fundamental solution is obtained, it is proved to be unique in a suitable space of distributions, and some of its regularity and integrability properties are described. The initial value problem for integrable and locally bounded initial data is then solved. Classical solutions are obtained as functions, whose regularity depends on time and that satisfy the expected conservation of energy. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Our purpose is to study the classical approximation of the linearized version of a three wave kinetic equation, around one of its equilibria, ∂u ∂τ (τ,x)= ∞ ˆ 0 (u(τ,y)−u(τ,x))K(x, y)dy, τ > 0,x>0 (1.1) E-mail address: [email protected]. https://doi.org/10.1016/j.jfa.2022.109390 0022-1236/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
2M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 K(x, y)=1 |x2−y2|−1 x2+y2y x,∀x>0,∀y>0,x=y. (1.2) In a condensed gas of quantum Bose particles ([17,27]), correlations arise between the superfluid component and the normal fluid part corresponding to the excitations. This causes number-changing processes, where an excitation splits into two others in presence of the condensate. A kinetic equation which includes these processes in a uniform Bose gas was first deduced in a series of papers by Kirkpatrick and Dorfman [22]. More recently, Zaremba & al. extended the treatment to a trapped Bose gas by including Hartree–Fock corrections to the energy of the excitations, and derived coupled kinetic equations for the distribution functions of the normal and superfluid components, sometimes called ZNG system (see [33]). Kinetic equations for quantum particles although similar in many aspects with the classical Botzmann equation, present new and interesting properties and have already been considered in the mathematical literature (cf. [14,23,29,30]and references therein). Only solutions that do not depend on the space variable are considered in this paper. First because our interest is mainly centered on the properties of the collision operator, but also because the homogeneity hypothesis simplifies very much the difficulties. These solutions are called spatially homogenous, or simply homogeneous. As noticed in [31], §5.2, they naturally arise in numerical analysis where all numerical schemes achieve a splitting of the transport operator and the collision operator. It is also expected that spatial homogeneity is a stable property, in the sense that a weakly inhomogeneous initial datum leads to a weakly inhomogeneous solution of the Boltzmann equation, as it has been mathematically justified in [2] under some ad hoc smallness assumptions. Under the conditions of spatial homogeneity, in the limit of temperature below but close to the critical temperature, the following system was first deduced in [10,22], ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂n ∂t (t, p)=C1,2(nc(t),n(t))(p)t>0,p∈R3,(a) n c(t)=−ˆ R3 C1,2(nc(t),n(t))(p)dp t > 0,(b) (1.3) where C1,2(nc, n)is the three waves collision integral, C1,2(nc(t),n(t)) = nc(t)I3(n(t))(p) (1.4) I3(n(t))(p)=¨ (R3)2R(p, p1,p 2)−R(p1,p,p 2)−R(p2,p 1,p)dp1dp2,(1.5) R(p, p1,p 2)= δ(|p|2−|p1|2−|p2|2)δ(p−p1−p2)× ×[n1n2(1 + n)−(1 + n1)(1 + n2)n].(1.6) In these notations n=n(t, p), n(t, p) denotes the density of particles in the normal gas that at time t >0have momentum pand energy ω(p) =|p|2, and nc(t)the density of
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 3 the condensate at time t. The term (1.4) describes the 1 ↔2 splitting of an excitation into two others in the presence of the condensate. For example (1.6)is for the splitting of the particle with momentum pin particles of momentum p1and p2, and similarly for R(p1, p, p2)and R(p2, p1, p). The specific form of such a term depends on the dispersion relation ω(|p|)for the energy of quasiparticles and on the matrix element Mof the effective Hamiltonian describing the interaction between them. The expression |p|2for the dispersion relation is deduced from the well established Bogoliubov approximation ([6], [17]) ω(|p|)=gn|p|2 m+|p|2 2m21/2 where mis the mass of the particles, g=4πam−1is the interaction coupling constant and ais the s-wave scattering length, nis the total particle density. When the temperature T of the gas is low but still such that kBT>>gn c(where kBis the Boltzmann constant) the approximations ω(|p|) ∼|p|2 2m+gncand |M|2=g2nc 2π2are used. In order to simplify the notations it is assumed in (1.4)–(1.6) without loss of generality, that the mass of the particles is m =2and the interaction coupling constant is g=1. Other theoretical models do exist to describe Bose gases in presence of a condensate (cf. [27]), but ZNG system, and (1.3a), (1.3b) in particular, are specially well suited to apply analytical PDE’s methods and obtain quantitative estimates of some important properties. It is well known that the equation (1.3a) has a family of non trivial equilibria n0, n0(p)=ν0(|p|2) (1.7) ν0(ω)=eβω −1−1,∀ω>0.(1.8) The parameter βmay be any positive constant and is related to the temperature T>0of the gas at equilibrium n0through the formula, β=1/(kBT) where kBis the Boltzmann’s constant. It is easily checked that R(p, pk, p) ≡0if n =n0. It is known (cf. [8]) that for all constants ρ >0and all non negative measures nin with a finite first moment, the system (1.3a)–(1.6)has a weak solution (n(t), nc(t)) with initial data (nin, ρ). For all t >0, n(t)is a non negative measure with finite first moment that does not charge the origin, and nc(t) >0. System (1.3a)-(1.6)was also treated in [3]. One basic aspect of the non equilibrium behavior of the system condensate–normal fluid is the growth of the condensate after its formation (cf. [33,5,27], and references therein). Although the relation of ncwith the condensate amplitude is not straightforward (cf. [33,19,27]), it seems nevertheless very closely related to the total number of particles of the system having an energy less than an arbitrarily small, but fixed, value (cf. [19]). It turns out that the evolution of nc(t) crucially depends on the behavior of
4M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 n(t, p)as |p| →0as indicated for example by Proposition 2 in [30]. When the measure n(t)is written as n(t, p) =|p|−1g(t, |p|2), it is proved in [8]that, if g(t)has no atomic part and has an algebraic behavior as |p| →0 then, n(t, p)= |p|→0a(t)|p|−2,(1.9) for some function a(t), (cf. [8]). These results of [8]and [30]make use of some regularity hypothesis on the solution n(t, p). But none of these properties have been proved to hold for the solutions nof the system (1.3a)–(1.6) obtained up to now. 1.1. Small isotropic perturbation of a Planck distribution Only radially symmetric perturbations of the equilibrium n0(p)are considered in what follows. Under such condition all the angular integrations can be performed in the collision integral (1.3a) and obtain an equation with only two real, non negative independent variables tand |p|. For non isotropic perturbations Ω(t, p), if expanded in spherical harmonics as Ω(t, p) =,m Ω,mY,m(p), similar, although slightly more involved equations are obtained for the evolution of the different angular momentum eigenstates Ω,m(t, |p|)(cf. equations (21), (22) in [16]). It would be of course of interest to know the possible effects of non radial perturbations, but this is out of the scope of this article and left for future work. In order to prove the existence of isotropic, regular classical solutions to (1.3a)-(1.6) satisfying (1.9), we first consider the linearization of (1.3a) around an equilibrium n0. The linearized equation was essentially obtained in [16]as briefly described in §5.3 of the Appendix: consider first the new isotropic dependent variable Ω, n(t, p)=n0(p)+n0(p)(1 + n0(p))Ω(t, |p|)=n0(p)+ Ω(t, |p|) 4sinh 2β|p|2 2.(1.10) When (1.10)is plugged in (1.3a), and only the linear terms in Ωare kept, then after the change of variables x=√β 2|p|,τ= t ˆ 0 mc0(s)π 32 β3 2 ds, u(τ,x)=Ω(t, |p|) |p|2,(1.11) the linearized equation for ureads (cf. [16]and §5.3 of the Appendix) ∂u ∂τ =pc(τ) ∞ ˆ 0 (u(τ,y)−u(τ,x))M(x, y)dy (1.12) M(x, y)=1 sinh |x2−y2|−1 sinh(x2+y2)y3sinh x2 x3sinh y2,(1.13)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 5 where pc(τ) =nc(t). When coupled with the equation p c(τ)=−pc(τ) ∞ ˆ 0 ∞ ˆ 0 (u(τ,y)−u(τ,x))M(x, y)x2dy dx, (1.14) it is easily checked that, if Fubini’s Theorem may be applied when the collision integral in (1.12)is multiplied by n0(x)(1 +n0(x))x2and n0(x)(1 +n0(x))x4and integrated over (0, ∞), p c(τ)+ d dτ ∞ ˆ 0 n0(x)(1 + n0(x))u(τ,x)x4dx =0, d dτ ∞ ˆ 0 n0(x)(1 + n0(x))u(t, x)x6dx =0. These identities reflect the conservation of the total number of particles and energy and therefore, system (1.12), (1.14) seems a reasonable linearization of (1.3a), (1.3b). The factor pc(τ)may now be scaled in equation (1.12)with a new change of time variable, denoted tagain with some abuse of notation, t= τ ˆ 0 nc(s)ds to obtain the equation, ∂u ∂t (t, x)= ∞ ˆ 0 (u(t, y)−u(t, x))M(x, y)dy (1.15) M(x, y)=1 sinh |x2−y2|−1 sinh(x2+y2)y3sinh x2 x3sinh y2.(1.16) The kernel Min (1.16) directly follows from the linearization of C1,2(nc(t), n(t)) in the right hand side of (1.3a) and the expression of n0(p)(1 +n0(p)) =eβ|p|2 22sinhβ|p|2 2−1 in the left hand side as explained in §5.3 of the Appendix. The Cauchy problem for equation (1.15)is still delicate and as a first step in that direction we consider in this work the simplified equation (1.1), (1.2), obtained only keeping in Mthe leading terms of the hyperbolic sine functions for small values of their arguments. This reminds somewhat the classical field limit were large occupation numbers of different modes are assumed ([27], Chapter 10).
6M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 For Kgiven in (1.2), our purpose is then to solve the following problem, ∂u ∂t (t, x)= ∞ ˆ 0 (u(t, y)−u(t, x))K(x, y)dy, t > 0,x>0,(1.17) L(u(t))(x)= ∞ ˆ 0 (u(t, y)−u(t, x))K(x, y)dy (1.18) u(0,x)=f0(x) (1.19) Again, for general, non necessarily isotropic, perturbations, similar simplified approximated equations may be obtained for the non radial components Ω,m of Ω(cf. equations (36), (38)–(43) in [16], for =1and =2). The Cauchy problem for equation (1.15)is considered [11]as a perturbation of (1.1). Integro differential equations of that form, in several dimensions but with integrals over all RN, have been much studied, under conditions on the kernels K, Mensuring that the integro differential operator satisfies an ellipticity property of some order s >0. The best known is the kernel C|x −y|−1−sfor s ∈(0, 2) that, for some constant C>0, gives the operator (−Δ)s/2. But weaker conditions on more general kernels may be found in [18]and the many references therein. A case where s =0is considered in [20]. For ua regular function, equation (1.1)may be written (cf (5.48)in the Appendix), ∂u ∂t (t, x)= ∞ ˆ 0 Hx y∂u ∂y(t, y)dy y(1.20) H(r)=10<r<1 1 rlog 1+r2 1−r2+1r>1 1 rlog 1−1 r4.(1.21) Equation (1.20)may be solved using the Mellin transform. Similar questions were considered with similar methods in [12], and in [13]for “post gelation” solutions of a coagulation equation. Some of the technical results in the last Section of [13] will be of some use in this work. The equation (1.3a) may actually be written as a coagulation-fragmentation equation, with nonlinear fragmentation, in terms of the energy ω=|p|2as independent variable for a measure gdefined as |p|n(t, p) =g(t, ω)(cf. [15], and [4]for general coagulation-collisional fragmentation equations). Remark 1.1. The linear equation (1.1)also follows if, first only quadratic terms are kept in (1.5), (1.6), and then linearization is performed around the equilibrium ω−1(p) =|p|−2. The first step yields a three wave turbulence type equation, considered by several authors [9,15,21], and (1.20)is the linearization of that equation around the equilibrium ω−1(p). Our setting is a bit narrow within the three waves area, since the specific form of the dispersion relation ωand of the matrix element Mare strongly used. Other examples
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 7 of three wave kinetic equations may be found in [32], for capillary waves, weak acoustic waves and others. 1.2. Main results The use of the Mellin transform, that is denoted by M, makes the spaces E p, q for p <q, presented for example in Chapter 11 of [24], very suitable. They are defined as the dual of the spaces Ep, q of all the functions φ ∈C∞(0, ∞)such that: Np,q,k(φ)=sup x>0kp,q(x)xk+1 φk(x)<∞ kp,q(x)=x−p,if 0 <x≤1 x−q,if x>1 with the topology defined by the set of seminorms {Np,q,k}k∈N. It follows that E p, q are the subspaces of D([0, ∞)) of Mellin transformable distributions ([24]). We call Sp,q ={s∈C;Re(s)∈(p, q)},∀p∈R,∀q∈R,p<q. (1.22) We also denote Hρ loc the set of locally Hölder continuous functions fof order ρthat satisfy, ∀K⊂(0,∞)compactset,∃CK>0; |f(x)−f(y)|≤CK|x−y|ρ,∀x∈K, ∀y∈K. For α∈(0, 1) and x >0we denote Θα(x) =|x −1|−α(log x); and for θ∈(0, 1), ||f0||1,θ =||f0||1+sup 0<x<1 xθ|f0(x)|. We denote arg and log the principal values of the argument and logarithm functions. The second moment of a function f(x), or M(f)(3), is sometimes called the energy of f, because it is equal, up to a constant, to the total energy of a system of particles with energy |p|2, whose momentum density function is n(p) =f(|p|). Theorem 1.2. There exists a function Λ ∈C((0, ∞); L1(0, ∞)) satisfying (1.20)) in D((0, ∞) ×(0, ∞)) and such that lim t→0Λ(t)=δ1,weakly in D(0,∞).(1.23) (log x)Λ ∈C((0,∞)×[0,∞)) (1.24) lim t→0t−1e−1/tY 1−2tΛt, 1+e−1/tY= 1 (1.25) uniformly for Yon bounded subsets of R. For all T>0, Λ(t) ∈E 0,2, M(Λ(t)) is bounded on S0,2for all t ∈(0, T). The function Λis such that,
8M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 (log x)∂mΛ ∂tm∈C((0,∞)×(0,∞)) ∀m∈N\{0},(1.26) (log x)2∂1+mΛ ∂tm∂x ∈C((0,∞)×(0,∞)),∀m∈N,(1.27) ∀k∈N,Λ∈Cmk+1 2,∞;Ck(0,∞),∀m∈N.(1.28) ∀r∈(0,1/2),∀α∈[0,r); ΘαΛ∈C(2r, 1) ; Hr−α loc (0,∞),(1.29) (where we recall that Θα(x) =|x −1|−αlog x), and satisfies (1.1)for almost every t >0 and x >0. The second moment of Λ(t)is one for all t >0. Theorem 1.3. If for some T>0, Λj∈C((0, T); L1(R+)), j=1, 2are supposed to be two solutions of (1.20), that satisfy (1.23), such that, for all t ∈(0, T), Λj(t) ∈E 0,2and M(Λ1(t) −Λ2(t)) is bounded in S0,2, then Λ1(t) =Λ 2(t)in E 0,2for all t ∈(0, T). The fundamental solution Λof the linearized equation inherits the conservation of the energy property that holds for the nonlinear equation (1.3a). As shown by (1.24), the Dirac measure at x =1is instantly regularized to a function Λ(t), whose regularity is given by (1.26)-(1.29). Property (1.25)shows that, for small values of t >0, Λ(t) behaves at x =1, like t|x −1|2t−1. The regularity of Λ(t)at x =1shown in Theorem 1.2 improves as the value of tincreases, as seen in (1.28). By (1.29), (3.54), for all t ∈(0, 1/2) the function Λ(t)is locally Hölder continuous around x =1of order r−αfor any r<2t and α∈(0, r). For t >1it follows from (1.28)that Λis C1. Detailed estimates of Λ(t, x) and some of its derivatives are given in the Sections below. This fundamental solution is used to solve the initial value problem. Theorem 1.4. Suppose that f0∈L1(0, ∞)and define, u(t, x)= ∞ ˆ 0 f0(y)Λ t y,x ydy y,∀t>0,∀x>0.(1.30) Then, u ∈L∞((0, ∞); L1(0, ∞)) ∩C((0, ∞); L1(0, ∞)) and it satisfies (1.20)in D((0, ∞) ×(0, ∞)), ∞ ˆ 0 u(t, x)x2dx = ∞ ˆ 0 f0(x)x2dx, ∀t>0,(1.31) there exists a constant C>0such that ||u(t)||1≤C||f0||1,∀t>0 (1.32) and u(t) t→0f0,in D(0,∞).(1.33)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 9 If f0∈L1(0, ∞) ∩L∞(0, ∞)then u(t) ∈L∞(0, ∞)for all t >0, there exists a constant C∞>0such that, ||u(t)||∞≤C∞||f0||∞,∀t>0.(1.34) If f0∈L1(0, ∞) ∩L∞ loc(0, ∞), L(u)∈L∞ loc((0,∞); L∞(0,∞)),(1.35) there exists a constant C>0such that, for all t >0and x >0, ∂u ∂t (t, x)≤⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Ct2x−4+t3x−3+ε||f0||1,a.e.x>3t C(1 + |log |x/2t−1||) (1 −θ)2xsup z∈(2t,3x)|f0(z)|, a.e.x ∈(2t/3,3t) C(t−2+t−3x)||f0||1,a.e.x∈(0,2t/3), (1.36) and usatisfies (1.1)for a.e. t >0, x >0. The solution ualso satisfies the following properties, Proposition 1.5. If f0∈L1(0, ∞)and uis given by (1.30), the following holds. 1.- For every δ>0as small as desired, and for all t >0, u(t, x)= x→0(f0,t)+⎛ ⎝t−2+δ t ˆ 0|f0(y)|dy +t5+δ ∞ ˆ t |f0(y)|dy y7⎞ ⎠Oδx1−δ,(1.37) (f0;t)=A1t−3 t ˆ 0 f0(y)y2dy +A2t−4 t ˆ 0 f0(y)y3dy + t ˆ 0 f0(y)b1t ydy y(1.38) for A1, A2constants given in (4.75)and b1(r) = r→∞ O(r−8)given in (3.15). 2.- For all t >0, the function u(t)is locally Hölder continuous on (0, ∞). More precisely, (i) There exist numerical constants C>0and σ∗ 0∈(−2, −1) such that |u(t, x)−u(t, x)|≤C||f0||1t−2+x−1−σ∗ 0t−1+σ∗ 0|x−x|,for 0<x <x<t, (1.39) (ii) For all c ∈(0, 2) there exists a constant C>0such that, |u(t, x)−u(t, x)|≤C||f0||1|x−x|x−1−ct−1+c+tx−4+ +C||f0||1|x−x|1−α x1−α|log(x/t)|1−α+|x−x|r−α txr−α|log(x/t)|(1+α)(r−α), if 0<x <t<x, (1.40)
16 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 where log(z) =log(|z|) +iArg(z)and Arg(z) ∈(−2π, 0]. The equation (2.1)on Vyields the following equation for H: ze−slog(−z)B(s)H(z,s)=−ze−slog(z)W(s−1)B(s−1)H(z,s −1) + 1 B(s)H(z,s)=−W(s−1)B(s−1)H(z,s −1) + eslog(−z) z B(s)H(z,s)=B(s)H(z,s −1) + eslog(−z) z and then, for all z∈Csuch that Re(z) >0and s ∈C, Re(s) ∈(0, 2) H(z,s)−H(z,s −1) = eslog(−z) zB(s).(2.23) We may use again the change of variables (2.9)and define, h(z,ζ)=H(z,s), B(ζ)=B(s) and deduce from (2.23)that hhas to satisfy h(z,r −i0) = h(z,r +i0) + e2iπβα(z)rα(z) z B(r),∀r>0; α(z)=log(−z) 2iπ .(2.24) It follows that α(z)=log(−z) 2iπ =−ilog |z| 2π+Arg(−z) 2π and the choice of the log(z)is such that −1 <(Re(α(z))) <0. By Proposition (2.5)it follows that the integral h(z,ζ)= 1 2iπ e2iπβα(z) z ∞ ˆ 0 rα(z) B(r) dr (r−ζ) is absolutely convergent and defines a function hanalytic on the domain {(z,s); z∈C,Re(z)>0,s∈C\[0,∞)} that satisfies (2.24). Using the original variables we obtain that H(z,s)=1 zˆ Re(σ)=β eσlog(−z) B(σ) dσ (1 −e2iπ(s−σ))(2.25) is well defined, analytic on z∈C, Re(z) >0, s ∈C, Re(s) ∈(β, β+1) where it satisfies
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 17 H(z,s)−H(z,s −1) = eslog(−z) zB(s).(2.26) Since β∈(0, 2) is arbitrary, using a contour deformation argument in the integral of the right hand side of (2.26), His extended as an analytic function z∈C, Re(z) >0and s ∈C, Re(s) ∈(0, 2). Using now (2.22)we recover the function V(z,s)=B(s) 2iπz ˆ Re(σ)=β e(σ−s)log(−z) B(σ) dσ (1 −e2iπ(s−σ)). Since Bis analytic and non zero on Re(s) ∈(0, 2) and β∈(0, 2) is arbitrary the function Vis analytic on z∈C, Re(z) >0and s ∈C, Re(s) ∈(0, 2) and satisfies the equation (2.21)for Re(s) ∈(1, 2). Corollary 2.9. The following inverse Laplace transform of V U(t, s)= 1 2iπ d+i∞ ˆ d−i∞ eztV(z,s)dz, β −1<d<β, is well defined for t >0and Re(s) ∈(0, 2). For all t >0, U(t, ·)is analytic on S0,2, ∀s∈S0,2,U(t, s)=B(s) 2iπ ˆ Re(σ)=β t−(σ−s)Γ(σ−s) B(σ)dσ, ∀β∈(Re(s),2) (2.27) may be extended to Cas a meromorphic function such that U(t, 3) =1and satisfies, ∀k∈N,U∈Ck((0,∞)×S0,2) (2.28) ∂U ∂t (t, s)=W(s−1)U(t, s −1) ∀t>0,∀s∈S1,3.(2.29) Proof. For all σand ssuch that Re(s) <Re(σ), and c >0, 1 2iπ c+i∞ ˆ c−i∞ ezt ze(σ−s)log(−z)dz =t−(σ−s)Γ(σ−s)e2iπ(σ−s)−1. We use now that Stirling’s formula for Γ(z)is uniformly valid for argz∈(−π+ε0, π−ε0) with ε0>0, to deduce that, for all R>0and β∈(0, 2) |Γ(σ−s)|≤CR e−π|σ| 2 1+|σ|,∀s;|s|≤R. (2.30)
18 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 The right hand side of (2.27)is then absolutely convergent. The identity (2.27)and the analyticity of U(t, ·)on S(0, 2) follow for β−1 <Re(s) <β. We also deduce from (2.30) that the integrals ˆ Re(σ)=β d dt t−(σ−s)Γ(σ−s) B(σ)dσ, k ∈N are absolutely convergent and analytic functions of son the strip Re(s) ∈(0, 2). Therefore, ∂k ∂tkU(t, s)=−B(s) 2iπ ˆ Re(σ)=β d dt t−(σ−s)Γ(σ−s) B(σ)dσ, and (2.28) follows. On the other hand, since 1 2iπ d+i∞ ˆ d−i∞ ezte(σ−s)log(−z)dz =t−(σ−s)−1Γ(1 + σ−s)e2iπ(σ−s)−1 the inverse Laplace transform of zV (z)is well defined for all t >0and given by, 1 2iπ d+i∞ ˆ d−i∞ eztzV (z,s)dz =−B(s) 2iπ ˆ Re(σ)=β t−(σ−s)−1Γ(1 + σ−s) B(σ)dσ. The expression (2.27) indicates that U(·, s) ∈C((0, ∞)). If the integration contour in (2.27)is deformed towards lower values of βand the pole of the function Γ(σ−s)at σ−s =0is crossed, U(t, s)=1−B(s) 2iπ ˆ Re(σ)=β t−(σ−s)Γ(σ−s) B(σ)dσ, β∈(0,Re(s)).(2.31) Since now Re(σ−s) <0, it follows that U(·, s) ∈C([0, ∞)) and U(0, s) =1. By classical deformation of contour arguments U(t)is now extended as a meromorphic function to all of C, and U(t, 3) =0by (2.31)and because B(3) =0(cf. Proposition 2.3). Using L(Ut(·,s)) (z)=zV (z,s)−U(0,s), we deduce ∂U ∂t (t, s)= 1 2iπ d+i∞ ˆ d−i∞ ezt (zV (z,s)−1) dz
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 19 We apply now the inverse Laplace transform to both sides of the equation (2.21)with Re(s) ∈(1, 2), since U(t)is analytic on S0,2and so is Won S−2,4, (2.29) follows. The following properties of Uare important for what follows, Proposition 2.10. For all T>0, there exists a positive constant CTsuch that for all s ∈S, t ∈(0, T), |U(t, s)|≤CTe−2tlog |bs|,b=eγe 2 2,(2.32) (1 + |s|) ∂U ∂s (t, s)+(1+|s|)2 ∂2U ∂s2(t, s)≤CTte−2tlog(|bs|)(2.33) ∂U ∂t (t, s)≤CTte−2tlog(|bs|)(1 + |log |s||) (2.34) (1 + |s|) ∂ ∂s ∂U ∂t (t, s)+(1+|s|)2 ∂2 ∂s2∂U ∂t (t, s)≤CT(1 + |log |s||) e2tlog(|bs|).(2.35) The proof of Proposition 2.10 is essentially the same as that of Proposition 8.1 in [13], only differing in small details, and is presented in the Appendix. It is based on the expression of U(t, s)given in (2.27)and also U(t, s)=−B(s) 2iπ ˆ Re(Y)=β−Re(s) t−YΓ(Y) B(s+Y)dY =ˆ Re(σ)=β eψ(s,σ,t)A(Y)dY (2.36) where Ψ(s, Y, t)= ˆ Re(ρ)=β log (−W(ρ)) Θ(ρ−s, Y )dρ −Ylog t−Y+Y−1 2log Y, with Θ defined in (2.19), and A(Y)= Γ(Y) 2iπe−YYY−1/2. The estimates for |s|follow from contour deformation methods on (2.27). The estimates for |s|large are deduced using the stationary phase argument on (2.36). As a Corollary, the inverse Mellin transform of U(t)is well defined. Corollary 2.11. For every t >0there exists a unique distribution Λ(t) := M−1(U(t)) ∈ E 0,2, the inverse Mellin transform of U(t)such that: M(Λ(t)) (s)=U(t, s),∀s∈S0,2(2.37) Λ∈C((0,∞); E 0,2).(2.38)
20 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 For all t >0it is given by the following expression, Λ(t, x)=x∂ ∂x2⎛ ⎝1 2πi c+i∞ ˆ c−i∞ U(t, s)s−2x−sds⎞ ⎠,c∈(0,2).(2.39) When t >1/2, Λ(t, x)= 1 2πi c+i∞ ˆ c−i∞ U(t, s)x−sds, c ∈(0,2).(2.40) Proof. By Corollary 2.9, for every t >0, the function U(t)is analytic on the strip Re(s) ∈(0, 2). By Proposition 2.10 |U(t, s)|≤|bs|−2t,∀t∈(0,1). It follows that, for all t >0, the function s−K+2U(t, s)is analytic and bounded on the strip Re(s) ∈(0, 2) as |s| →∞for K=2. It follows from Theorem 11.10.1 in [24]that there exists a unique tempered distribution Λ(t) ∈E 0,2that satisfies (2.37)and is given by (2.39). As soon as t >1/2, the integral in the right hand side of (2.40)is absolutely convergent and its Mellin transform is U(t)from where it is equal to Λ(t). Property (2.38) follows from (2.28)and the continuity of the inverse Mellin transform. It is now possible to apply the inverse Mellin transform to both sides of (2.29). Proposition 2.12. Λ(t)∈C1(0,∞;E 1,3) (2.41) ∂Λ ∂t =∂Λ ∂x ∗Hin C((0,∞); E 1,3) (2.42) where His the function defined in (1.21). Proof. By (2.38), ∂xΛ(t) ∈C(0, ∞; E 1,3)and for all s ∈S1,3, M(∂xΛ(t))(s)=−(s−1)U(s−1),and Since M(H)(s) =−W(s−1) s−1, it then follows for all t >0, M−1(W(s−1)U(t, s −1))(x)=∂Λ(t) ∂x ∗H(x)inE 1,3 On the other hand, by (2.29)and Proposition 2.10
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 21 M−1∂U(t) ∂t (x)≡M−1(W(s−1)U(t, s −1)) = =x∂ ∂x2⎛ ⎝1 2πi c+i∞ ˆ c−i∞ W(s−1)U(t, s −1)s−2x−sds⎞ ⎠.(2.43) By Proposition 2.10 again, for all t >0and x >0, d dt ⎛ ⎝1 2πi c+i∞ ˆ c−i∞ U(t, s)s−2x−sds⎞ ⎠=1 2πi c+i∞ ˆ c−i∞ W(s−1)U(t, s −1)s−2x−sds (2.44) and the integral in the right hand side of (2.44)is absolutely convergent, uniformly for xand tin compacts subsets of (0, ∞) ×(0, ∞). It is then a continuous function on (0, ∞) ×(0, ∞). It is then possible to apply the operator (x∂x)2to both sides of (2.44) in the sense of distributions to obtain (2.42). The following Proposition, shows some important properties of Λ. Proposition 2.13. The function Λ(t)defined in Corollary 2.11 satisfies (1.24), and (1.26)–(1.29). Proof. By its definition, Λ(t) ∈E 0,2, and for all m ∈N, M(((log x)∂m tΛ(t))(x)) = ∂s∂m tU(t, s) =∂sU(t, s −m) m =1 W(s−)in Sm,2+m.(2.45) Property (1.26) follows from the decay of the function at the righthand side of (2.45)as |Im(s)| →∞. Indeed, by Proposition 2.10, (2.4)and (2.5), for every m ≥1and t >0 there exists a positive constant C>0, depending on m, band t, such that, ∂sU(t, s −m) m =1 W(s−)≤C(1 + |s|)−1−t,∀s∈Sm,2+m. It follows that, for c∈(m, 2 +m), ((log x)∂m tΛ(t))(x)= 1 2πi c+i∞ ˆ c−i∞ ∂sU(t, s −m) m =1 W(s−)x−sds (2.46) where the integral in (2.46)is absolutely convergent. Since moreover the integral converges uniformly for xand ton compact subsets of (0, ∞) ×(0, ∞), property (1.26) follows. Notice that the same proof shows (log x)Λ ∈C((0, ∞) ×(0, ∞)). Similarly,
22 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 M(((log x)2∂x∂m tΛ(t))(x)) = ∂2 s(log(s−1) ∂m tU(t, s)) =∂2 slog(s−1) U(t, s −m) m =1 W(s−),in Sm,2+m. (2.47) Again by Proposition 2.10, (2.4)and (2.5), for every m ≥0, t >0, there exists a positive constant C>0such that, ∂2 sU(t, s −m) m =1 W(s−)≤C(1 + |s|)−2−t|log s)|,∀s∈Sm,2+m. Then, (log x)2∂m t(t, x)(t, x)= 1 2πi c+i∞ ˆ c−i∞ ∂2 sU(t, s −m) m =1 W(s−)x−sds and, since the integral is absolutely and uniformly convergent for xand ton compact subsets of (0, ∞) ×(0, ∞), property (1.27) follows. In order to prove (1.24)we first notice that for t >1/2, formula (2.40)may be used. Using (3.4), if we deform the integration contour in (2.40) towards lower values of Res and cross the pole of B(s)at s =0, using Res(B(s), s =0)) =−B(1)/W (0) we obtain Λ(t, x)= 1 4π2ˆ Re(s)=c x−sˆ Re(σ)=β B(s) B(σ)Γ(σ−s)t−(σ−s)dσds =B(1) 2iπW (0) ˆ Re(σ)=β Γ(σ)t−σ−1 B(σ)dσ+ +1 4π2ˆ Re(s)=c x−sˆ Re(σ)=β B(s) B(σ)Γ(σ−s)t−(σ−s)dσds, c ∈(−1,0) It follows that Λ ∈C([1/2, ∞) ×[0, ∞)) since both integrals converge uniformly for x and ton compact subsets of [0, ∞) ×[1/2, ∞). For t ∈(0, 1/2) (log x)Λ(t, x)= 1 2πi c+i∞ ˆ c−i∞ ∂sU(t, s)x−sds. (2.48) It follows from (2.27)that U(t)is meromorphic on the strip S−1,2with a simple pole at s =0. Then, ∂sU(t, s)is also meromorphic on S−1,2and has a pole of order 2 at s =0. We deduce
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 23 (log x)Λ(t, x)=−B(1) 2iπW (0) ˆ Re(σ)=β Γ(σ)t−σ−1 B(σ)dσ +1 2πi c+i∞ ˆ c−i∞ ∂sU(t, s)x−sds, for c ∈(−1, 0) and, arguing as before, (log x)Λ ∈C((0, 1/2) ×[0, ∞)) and (1.24) follows. In order to prove (1.28)we notice that, by Proposition 2.10, (2.4)and (2.5) again, for all k∈Nand m ∈Nthere exists C>0such that (s−k)kU(t, s −m) m =1 W(s−)≤C|s|k−2t|log |s||m,for |s|>> 1. Then, for t >1, k<2t −1, and c∈(m +k, 2 +m +k)the identity (2.40)may be used to write ∂k+mΛ ∂xk∂tm=(−1)k 2πi c+i∞ ˆ c−i∞ (s−k)kU(t, s −m) m =1 W(s−)x−s−kds, (2.49) where the integral in (2.49)converges absolutely. Since the convergence is uniform in compacts of ((k+1)/2, ∞) ×(0, ∞), property (1.28) follows. We prove now property (1.29). For all t ∈(0, 1/2), r∈(0, 2t), and |s|large, ∂ ∂sU(t, s −r)Γ(1 −s+r) Γ(1 −s)x1−s≤|x|−Re(s+r)|s|−2t−1+r,(2.50) the fractional derivative of order rof (log x)Λ, is then ∂r(log x)Λ(t) ∂xr=1 2πi c+i∞ ˆ c−i∞ Γ(1 −s+r) Γ(1 −s) ∂ ∂sU(t, s −r)x−sds (2.51) c∈(r, 2), (cf. [26], §2.10), where the integral in the right hand side of (2.51)converges absolutely for xand tin compact subsets of (0, ∞) ×(0, ∞). For each t >0the function (log x)Λ(t)has continuous fractional x-derivative of order ron every compact subset of (0, ∞)and by (2.51), for all t >0 ∀r∈(r, 2),∃Cr>0, ∂r((log x)Λ(t, x)) ∂xr≤Crx−r,∀x>0.(2.52) By Theorem 3.1 [28], (1.29) follows for α=0and r∈(0, 2t). Moreover, since (log x)Λ(t, x)= 1 2iπ ˆ Re(s)=c ∂U(t, s) ∂s x−sds,
24 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 by the continuity property (1.26), and an integration by parts, lim x→1(log x)Λ(t, x)= 1 2iπ ˆ Re(s)=c ∂U(t, s) ∂s ds =0. Then property (1.29)is deduced for α∈(0, r)using the result in [25], p. 14. Corollary 2.14. The function Λsatisfies lim t→0Λ(t)=δ1,in D(0,∞).(2.53) Proof. Consider any test function ϕ ∈D(0, ∞)and suppose that supp(ϕ) ⊂(a, b)for some 0 <a <b <∞. Then Λ(t),ϕ−ϕ(1) = ∞ ˆ 0 M−1(U(t)−1) (x)ϕ(x)dx =1 2iπ ∞ ˆ 0 c+i∞ ˆ c−i∞ (U(t, s)−1) x−sdsϕ(x)dx =1 2iπ c+i∞ ˆ c−i∞ ∞ ˆ 0 x−sϕ(x)dx (U(t, s)−1) ds =1 2iπ c+i∞ ˆ c−i∞ M(ϕ)(1 −s)(U(t, s)−1) ds. By definition, for s =c +iv, v∈R, Re(s) ∈(β, 2), M(ϕ)(1 −s)= ∞ ˆ 0 ϕ(x)x−sdx =1 (1 −s)(2 −s) ∞ ˆ 0 ϕ(x)x2−sdx ≤C 1+|s|2. As we have seen above (cf. (2.31)), for Re(s) ∈(β, 2), |U(t, s)−1|=|B(s)|ˆ Re(σ)=β t−(σ−s)Γ(σ−s) B(σ)dσ ,β ∈(0,Re(s)) ≤|B(s)|tRe(s−β)ˆ Re(σ)=β t−(iIm(σ−s))Γ(σ−s) B(σ)dσ ≤CeRe(s−β)logtlog |s|
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 25 Then, for Re(s) =c >β : |Λ(t),ϕ−ϕ(1)|=1 2iπ c+i∞ ˆ c−i∞ M(ϕ)(1 −s)(U(t, s)−1) ds ≤Ce(c−β)logt c+i∞ ˆ c−i∞ |M(ϕ)(1 −s)|log |s||ds|≤Ce(c−β)logtˆ R log |v|dv 1+|v|2→ t→00. 3. Further properties of Λ In this Section we first give the main terms in the asymptotic behavior of the fundamental solution Λ(t, x)in different regions of the t, x)plane. More detailed results are also given on the continuity and derivability properties of the function Λ, in particular around the point x =1, where the Dirac’s delta formation is described. These results are used later, first to solve the Cauchy problem associated to equation (1.1)for a large set of initial data, and then to get the precise behavior of the solutions. This will be mainly done with the representation of Λas a contour integral, using the classical contour deformation argument and Cauchy’s residue Theorem. ρ(σ)=Res1 B(s),s=σ,r(σ)=Res(B(s),s=σ) (3.1) ˜r(σ)=Res(s−2B(s),s=σ),˜ρ(σ)=Res1 W(s),s=σ(3.2) P(n)=ResΓ(ω) B(ω),ω=−n,Q(n)=ResΓ(ω+1) B(ω),ω=−n=−nP(n).(3.3) Notice that −nis a simple pole of Γ(ω) B(ω)for n ∈{0, ···5}and a double pole for n ≥6. 3.1. Behavior of Λfor t >1 The function Λsatisfies the following estimates when t >1 Proposition 3.1. For all t >1, Λ(t, x)=t−3Q1(θ)+Q2(t, θ),θ=x t(3.4) Q1(θ)= c1 2iπ ˆ Re(s)=c θ−sB(s)Γ(3 −s)ds (3.5) Q2(t, θ)=−1 4π2ˆ Re(s)=c θ−sˆ Re(σ)=β2 B(s) B(σ)Γ(σ−s)t−σdσds (3.6)
32 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 since the first pole of Γ(σ−σ∗ n)/B(σ)with negative real part is located at σ=−6, (3.28) follows. The same method gives estimate (3.29). Starting from (2.39)and (2.29), we deduce ∂ ∂tΛ(t, x)=x∂ ∂x2⎛ ⎝1 2πi c+i∞ ˆ c−i∞ U(t, s −1)W(s−1)s−2x−sds⎞ ⎠ =−x∂ ∂x2⎛ ⎜ ⎝x−1 4π2 c+i∞ ˆ c−i∞ ˆ Re(σ)=β B(s)Γ(σ−s+1) B(σ)s−2t−σx t−(s−1) dσds⎞ ⎟ ⎠. With the same argument as before we deduce, ∂ ∂tΛ(t, x)=x−1μ(t) ∞ k=1 x t−k−β 1+1 Ak(k+β 1)2+x−1 ∞ n=1 x t−4n(4n+1) 2νn(t) ∂ ∂tΛ(t, x)≤C1x−1x t−β 1t6+C2x−1x t−4t2,x t>1,0<t<1. For estimate (3.30), where x ∈(0, t/2) the sintegration contour is moved towards smaller values of Re(s). The sequence of poles of B(s), with Re(s) ≤0is then crossed. These are located at s =0, −1and points σ∗ nof Proposition (2.1). We deduce, arguing as before ∂tΛ(t, x)=x∂ ∂x21 t˜μ1(t)+˜μ2(t)x t2+ ∞ n=0 x t−σ∗ n˜νn(t), =˜μ2(t)x t2+ ∞ n=0 (σ∗ n)2x t−σ∗ n˜νn(t) ˜νn(t)= ˜rσ∗ n 2iπ ˆ Re(ω)=β Γ(ω−σ∗ n) B(ω)t−ωdω;˜μ2(t)= ˜r−1 2iπ ˆ Re(σ)=β Γ(σ+2) B(σ)t−σdσ The functions ˜νnand ˜μ2are now determined by the sequence of zeros of B(σ)such that Re(σ) ≤0. Since the first one is at s = 6 estimate (3.30) follows. From (2.39)and basic properties of the Mellin transform, ∂Λ ∂x(t, x)=x∂ ∂x3 (J(t, x)) where, for c∈(1,2), J(t, x)=−1 4π2 c+i∞ ˆ c−i∞ ˆ Re(σ)=β t−σ−1B(s−1)Γ(σ−s) B(σ)(s−1)s−3x t−sdσds.
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 33 For t ∈(0, 1) and x ∈(0, ρt)the behavior of J(t, x)is obtained by deforming the contour integrals to lower values, first of Re(s)and then of Re(σ). In the first step we cross first the pole s =0then, the poles σ∗ n+1of B(s −1) of (s −1)−2B(s), and obtain, J(t, x)=ResB(s−1) s3;s=0 R∗ 0(t)+ 2 j=1 x t−1−σ∗ j R∗ j(t)+Ox−1−σ∗ 3+εt−σ∗ 3−ε R∗ 0(t)= ˆ Re(σ)=β t−σ−1Γ(σ) B(σ)dσ, R∗ j(t)=r(1 + σ∗ j)σ∗ j (1 + σ∗ j)3−t−σ∗ j B(1 + σ∗ j)+Ot1−σ∗ j and, (3.33) follows, with the same argument as in the proof of (3.27), (3.28). The estimate (3.34) where x/t >x >1 requires to deform first the scontour integrals in Jtowards larger values of Re(s). Since by construction c <βwe first the pole of Γ(σ−s)at s =σ, from where, for c∈(β, 2), J(t, x)=t−1 2iπ ˆ Re(σ)=β (σ−1) W(σ−1)σ−3x−σdσ +J1(t, x) J1(t, x)=−1 4π2ˆ Re(σ)=β c+i∞ ˆ c−i∞ t−σ−1(s−1)B(s−1)Γ(σ−s) B(σ)s3x t−sdσds J(t, x)=2˜ρ(2) 27 t−1x−3+Ot−1x−4+ε−J1(t, x),for arbitrarily small ε>0 The sintegration contour in J1is moved to larger values. The next pole of B(s −1) is at s =6. Since σ=3is a zero of B(σ), that we do not want to cross, the condition σ−s ∈(−1, 0) can not be maintained. The singularities of Γ(σ−s)at σ−s =−1and σ−s =−2are crossed. Since 1 2iπ ˆ Re(σ)=β x−σ−1σ (σ+1) 3dσ =1 2(log x)21(0,1)(x)=0,∀x>1, this gives, using (2.15), for d ∈(5, 6), J1(t, x)=−t 2iπ ˆ Re(σ)=β x−σ−2(σ+1) (σ+2) 3W(σ)dσ +J2(t, x) J2(t, x)= 1 4π2ˆ Re(σ)=β d+i∞ ˆ d−i∞ t−σ−1(s−1)B(s−1)Γ(σ−s) B(σ)s3x t−sdσds then J1(t, x)=O(tx−6)+J2(t, x),J 2(t, x)=O(t1+εx−5−ε) and (3.34) follows from the location of the zeros and poles of Wand B.
34 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 3.2.2. Properties of Λfor t ∈(0, 1) and 0 <|x −1| <1 Proposition 3.6. There exists a constant Csuch that |Λ(t, x)|≤ Ct |x−1|,∀x;0<|1−x|<1,∀t∈(0,1) (3.47) ∂ ∂tΛ(t, x)≤C(1 + t|log |x−1||) |x−1|,∀x;0<|1−x|<1,∀t∈(0,1).(3.48) Proof. We define the new variables X=logx, ˜ Λ(t, X)=Λ(t, x),∀t>0,x>0.(3.49) Then, ∀X∈R,˜ Λ(t, X)= 1 2iπ ˆ Re(s)=c e−sXU(t, s)ds. (3.50) After two integrations by parts: ˜ Λ(t, X)= 1 X2ˆ Re(s)=ce−sX −1∂2U ∂s2(t, s)ds. (3.51) When |s| <1, we use |e−sX −1| ≤|sX|and deduce from (3.51)and Proposition 2.10 ˜ Λ(t, X)≤t |X|ˆ Re(s)=c |s|<1 |s||ds| 1+|s|2+ 1 X2ˆ Re(s)=c |s|>1e−sX −1∂2U ∂s2(t, s)ds . But, ˆ Re(s)=c |s|>1e−sX −1∂2U ∂s2(t, s)ds =1 Xˆ Re(u)=cX |u|>|X| e−u−1∂2U ∂s2t, u Xdu and by Proposition 2.10 ˆ Re(s)=c |s|>1 e−sX −1 1+|s|2ds ≤t |X|ˆ Re(u)=cX |u|>|X| |e−u−1| 1+|u/X|2|du| =t|X|ˆ Re(u)=cX |u|>|X| |e−u−1| |X|2+|u|2|du|<t|X|ˆ Re(u)=cX |u|>|X| |e−u−1| |u|2|du|.
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 35 If s =c +iv, then e−u=e−cXe−iv, and for Xbounded, e−u−12=e−2cX (cos2(vX)−1) + sin2(vX)≤C and, if u =cX +iw, ˆ Re(u)=cX |u|>|X| |e−u−1| |u|2|du|≤Cˆ Re(u)=cX c2|X|2+w2>|X|2 dw c2X2+w2≤Cˆ R dw c2X2+w2=C X This shows (3.47)and a similar calculation gives (3.48)using that, ∀X∈R,∂ ∂t ˜ Λ(t, X)= 1 2iπ ˆ Re(s)=c e−sX ∂U ∂t (t, s)ds. Lemma 3.7. For all ε >0as small as desired, there exists a constant Cε>0such that for all t ∈(0, 1), α∈(0, 2t), and all x ∈(0, 2), |(log x)1−αΛ(t, x)|≤ Cε xε|log x|α.(3.52) Proof. It follows from (2.33)that, there exists C>0, independent of ε, such that for t ∈(0, 1) and x ∈[0, 2), |(log x)Λ(t, x)|≤Ct ˆ Re(s)=ε (1 + |s|)−1−2tx−sd|s|=Cε xε, and then, for all α∈(0, 1), |(log x)1−αΛ(t, x)|=|(log x)Λ(t, x)| |log x|α≤Cεx−ε |log x|α. Our next goal is an estimate of the Hölder property (1.29) for Λ(t). We start with, Lemma 3.8. For all r∈(0, 1/2), all ε >0arbitrarily small, there exists a constant C(r, ε) >0such that, for α∈[0, r)and a, bsatisfying 0 <a <b <∞, ∀t∈(0,1),∀(x, y)∈(a, b)×(a, b), |Θα(x)Λ(t, x)−Θα(y)Λ(t, y)|≤(2 + α)C(r, ε)|x−y|r−α aε(3.53) where, Θα=|x −1|−α(log x)
36 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 Proof. We deduce from (1.24)and (3.52)(cf. Theorem 3.1 [28]for example) that for all ε >0 arbitrarily small, there exists a constant C(r, ε) >0such that, for a, bsatisfying 0 <a <b <∞, ∀t∈(0,1),∀(x, y)∈(a, b)×(a, b), |(log x)Λ(t, x)−(log y)Λ(t, y)|≤C(r, ε)|x−y|r aε This is (3.53)for α=0. Property (3.53)for all α∈[0, r) follows by simple straightforward calculation (cf. for example 5oin [25], p. 14). Corollary 3.9. For all r∈(0, 1/2), for all ε >0arbitrarily small, there exists a constant C=C(r, α) >0such that, for α∈[0, r)and a ∈(0, 2), ∀t∈(0,1),∀(x, y)∈(a, 2) ×(a, 2), |log x|1−αΛ(t, x)−|log y|1−αΛ(t, y)≤C|x−y|r−α ar−α|log a|(1+α)(r−α)(3.54) Proof. Let us write, |log x|1−αΛ(t, x)=ϕ(t, x)w(x) ϕ(t, x)=(log x)Λ(t, x) |x−1|α,w(x)=|x−1|α log x|log x|1−α. By Lemma 3.8 and the mean value Theorem, ϕ(t, x)w(x)−ϕ(t, y)w(y)=(ϕ(t, x)−ϕ(t, y))w(y)+ϕ(x)(w(t, x)−w(t, y)) |ϕ(t, x)−ϕ(t, y)||w(y)|≤(2 + α)C(r, T, ε)|x−y|r−α aεsup z∈(a,2) |w(z)| |w(x)−w(y)|≤ sup z∈(0,2) |w(z)||x−y|≤ C a|log a|1+α|x−y|, because, w(x)=α|x−1|α−2(x−1) log x|log x|1−α−|x−1|α x|log x|2|log x|1−α+ +|x−1|α log x|log x|−αH(x−1) x,H= Heaviside’s function |w(x)|≤ C a|log a|1+α,∀x∈(a, 2). But since, |w(x)| ≤w(2) for all x ∈(0, 2) we deduce by interpolation, for all θ∈(0, 1),
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 37 |w(x)−w(y)|≤ C|x−y|θ aθ|log a|(1+α)θ and, for θ=r−αthat may be supposed to be larger that ε, |ϕ(t, x)w(x)−ϕ(t, y)w(y)|≤(4 + 2α)C(r, ε)|x−y|r−α aε+C|x−y|r−α ar−α|log a|(1+α)(r−α) ≤C|x−y|r−α ar−α|log a|(1+α)(r−α) for some constant C>0that depends on r, αbut not on a. Proposition 3.10. For all r∈(0, 1/2), α∈[0, r)and t ∈(2r, 1), if m(x, y) =min(x, y) |Λ(t, y)−Λ(t, x)|≤ 2|Λ(t, x)||x−y|1−α m(x, y)1−α|log y|1−α+C|x−y|r−α m(x, y)r−α|log m(x, y)|(1+α)(r−α)(3.55) and the function Λsatisfies property (1.29) Proof. Λ(t, y)−Λ(t, x)=|log x|1−αΛ(t, x)A1(x, y)+A2(t, x, y) (3.56) A1(x, y)=1 |log x|1−α−1 |log y|1−α(3.57) A2(t, x, y)=|log x|1−αΛ(t, x)−log y)1−αΛ(t, y) |log y|1−α(3.58) |A1(x, y)|=|log x|1−α−|log y|1−α |log x|1−α|log y|1−α≤||log x|−|log y||1−α |log x|1−α|log y|1−α =1 |log x|1−α|log y|1−αd dz |log z|(ξ)|x−y|1−α =1(1,∞)(ξ)|x−y|1−α ξ1−α|log x|1−α|log y|1−α for some ξbetween xand y, and then, |A1(x, y)|≤ 2|x−y|1−α min(x, y)1−α|log x|1−α|log y|1−α.(3.59) Using Corollary 3.9 to estimate A2, the result follows. 3.3. Behavior of Λas x →1 In the following Proposition the behavior of Λis given in the neighborhood of x =1. Its proof, somewhat technical is given in the Appendix.
38 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 Proposition 3.11. For all bounded subset A ⊂R, there exists a constant Ca>0such that, sup X∈A, t∈(0,1) t−1|X|1−2t|˜ Λ(t, X)|≤CA(3.60) sup X∈A, t∈(0,1) |X|1−2t (1 + 2tlog |X|) ∂˜ Λ ∂t (t, X)≤CA,(3.61) and, uniformly on A, lim t→0t−1|X|1−2t˜ Λ(t, X)=1,(3.62) lim t→0|X|1−2t (1 + 2tlog |X|) ∂˜ Λ ∂t (t, X)=1.(3.63) Remark 3.12. For any ϕ ∈CC(R), lim t→0tˆ R |X|−1+2tϕ(X)dX =ϕ(0). Corollary 3.13. lim t→0t−1e−1/tY 1−2tΛt, 1+e−1/tY= 1 (3.64) uniformly on bounded subsets of R. Proof. For t >0 sufficiently small, depending on the bounded set Kof Rwhere Yvaries, 1 +e−1/tY>0. Then we define 1 +e−1/tY=eXand by definition Λ(t, 1 +e−1/tY) = ˜ Λ(t, X). By (3.62), uniformly for Xin bounded subsets of R, lim t→0t−1|X|2t−1˜ Λ(t, X) = 1 (3.65) lim t→0t−1|X|2t−1Λ(t, 1+e−1/tY) = 1 (3.66) But, since lim t→0e−1/tY=0,uniformly for Yon K, it follows that lim t→0eX=1,uniformly for Yon K. Then
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 39 lim t→0 e−1/tY X= lim t→0 eX−1 X=1 from where lim t→0t−1|X|2t−1Λ(t, 1+e−1/tY) = lim t→0t−1|e−1/tY|2t−1Λ(t, 1+e−1/tY) = 1 (3.67) uniformly for Y∈K. Corollary 3.14. For all R∈(0, 1) there exists CR>0such that |Λ(t, x)|≤ CRt |x−1|1−2t,∀x;|x−1|e1/t ≤R, t ∈(0,1),(3.68) ∂ ∂tΛ(t, x)≤CR(1 + 2tlog |x−1|) |x−1|1−2t,∀x;|x−1|e1/t ≤R, t ∈(0,1).(3.69) Proof. By (3.60), for all t ∈(0, 1), |Λt, 1+e−1/tY|≤ 2t e−1/tY1−2t,∀Y∈(−R, R).(3.70) In terms of x =1 +e−1/tY, (3.68) follows. Similarly, (3.69) follows from (3.61). Corollary 3.15. The function Λsatisfies, Λ∈C((0,∞),L 1(0,∞)),(3.71) and there exists C>0such that, ||Λ(t)||1≤C 1+t2,∀t>0 (3.72) Proof. We prove (3.72) first. For t ∈(0, 1) we use the estimates in Section 3.2 ∞ ˆ 0|Λ(t, x)|dx = 1/2 ˆ 0|Λ(t, x)|dx +ˆ |x−1|<1/2 |Λ(t, x)|dx + ∞ ˆ 3/2 |Λ(t, x)|dx. (3.73) The first and third integrals in the right hand side of (3.73)are estimated as, 1/2 ˆ 0|Λ(t, x)|dx ≤t 1/2 ˆ 0 dx |x−1|≤t. ∞ ˆ 3/2 |Λ(t, x)|dx ≤C1t7+β 1 ∞ ˆ 3/2 x−1−β 1dx +C2t7 ∞ ˆ 3/2 x−6dx.
40 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 For the second integral in the right hand side of (3.73)we write, ˆ |x−1|<1/2 |Λ(t, x)|dx =ˆ 0<|x−1|<e−1/t |Λ(t, x)|dx +ˆ e−1/t<|x−1|<1/2 |Λ(t, x)| ≤Ct ˆ 0<|x−1|<e−1/t dx |x−1|1−t+Ct ˆ e−1/t<|x−1|<1/2 dx |x−1| =2Ct e−1/t ˆ 0 dz z1−t+2Ct 1/2 ˆ e−1/t dz z=2C e−2Ctlog 2 + 2C. For t >1, by Proposition (3.1) ∞ ˆ 0|Λ(t, x)|dx =t−3 ∞ ˆ 0|Q1(θ)|dx + ∞ ˆ 0|Q2(t, θ)|dx, =t−2 ∞ ˆ 0|Q1(θ)|dθ +t ∞ ˆ 0|Q2(t, θ)|dθ, where we used the change of variable θ=x t. Then (3.72) follows since, by Proposition (3.2), Q1∈L1(0, ∞) and, by Proposition (3.3), ∞ ˆ 0|Q2(t, θ)|dθ ≤Ct−4(3.74) On the other hand if t1>0and |t −t1| <t 1/4, for any ε >0small fixed and Rlarge to be fixed, ∞ ˆ 0|Λ(t1,x)−Λ(t, x)|=I1+I2+I3+I4 I1= 1−ε ˆ 0|Λ(t1,x)−Λ(t, x)|dx ≤sup x∈[0,1−ε)|Λ(t1,x)−Λ(t, x)| I2= 1+ε ˆ 1−ε|Λ(t1,x)−Λ(t2,x)|dx ≤2εsup x∈[1−ε,1+ε) t∈3t1 4 5t1 4|Λ(t, x)| I3= R ˆ 1+ε|Λ(t1,x)−Λ(t2,x)|dx ≤sup x∈[1+ε,R)|Λ(t1,x)−Λ(t, x)|
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 41 I4= ∞ ˆ R |Λ(t1,x)−Λ(t2,x)|dx ≤ ∞ ˆ R |Λ(t1,x)|dx + ∞ ˆ R |Λ(t2,x)|dx The terms I1, I2and I3tend to zero as t →t1by the continuity of (log x)Λ(t, x)for t >0and x ∈R+\{1}. If 0 <t 1<1, we deduce I4≤CR−β 1from an estimate similar to (3.71) written for Rinstead of 3/2. For t >1, it follows from (3.4)and (2.30)that I4≤CR1−cwhere cmay by chosen in the interval (0, 2). The choice c ∈(1, 2) ensures that for all t >0, I4→0when R→∞. This proves (3.71). In order to check that Λsatisfies (1.1)let us show first that L(Λ(t)) is well defined. When t >1this follows from the C1regularity of the function Λ(t). Proposition 3.16. L(Λ) ∈C((1, ∞) ×(0, ∞)). For all t >1, there exists a numerical constant C>0such that L(Λ(t))(x)<C xt2min 1 t,1 x,∀x>0. Proof. For t >1, Λ(t) ∈C1(0, ∞)and by Propositions 3.1–3.3 |Λ(t, x)|≤min(t−3,x −3). Therefore, for every x >0, and y∈(0, x/2) |Λ(t, y)−Λ(t, x)|K(x, y)≤Cx−2min(t−3,x −3)+min(t−3,y−3) Then, if x ∈(x0−ε, x0+ε)for some x0>2ε >0, |Λ(t, y)−Λ(t, x)|K(x, y)10<y<x/2≤C(x0−ε)210<y<(x0+ε)/2 (min(t−3,(x0−ε)−3)+min(t−3,y−3)) and since the right hand side belongs to L1(0, ∞)it follows that x/2 ˆ 0 (Λ(t, y)−Λ(t, x))K(x, y)dy ∈C(0,∞). Moreover: x/2 ˆ 0|Λ(t, y)−Λ(t, x)|K(x, y)dy ≤Cmin(t−3,x −3)x−1+ +Cx−2 x/2 ˆ 0 min(t−3,y−3)dy ≤Cmin(t−3,x −3)x−1+C xt2min(t−1,x −1).
48 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 ∂U ∂t (t, s)=W(s−1)U(t, s −1) + r(t, s) (3.87) r(t, s)=M(Λ(t))(s)(t) (3.88) and the function ris bounded on (0, T) ×S0,2, r(t) ≡0if 0 ≤t ≤T/2. We may then Laplace transform both sides of (3.87)and obtain, for some constant C>0, z˜ V(z,s)=−W(s−1) ˜ V(z,s −1) + ˜r(z,s),Re(z)>0,Re(s)∈(1,2) (3.89) |˜r(z,s)|≤Ce−T 2Re(z),∀s∈S,Re(z)>0.(3.90) The function ˜ Vmay be split as ˜ V=˜ Vp+˜ Vhwhere ˜ Vpis the particular solution of (3.89), ˜ Vp(z,s)= 1 2iπ B(s) zˆ Re(σ)=β e(σ−s)log(−z) B(σ) ˜r(z,σ)dσ (1 −e2iπ(s−σ)) and ˜ Vhmust satisfy ∂˜ Vh ∂t (t, s)=−W(s−1) ˜ Vh(t, s −1),Re(z)>0,Re(s)∈(1,2) (3.91) The function ˜ Vp(z, s)is analytic on s ∈Sfor all Re(z) >0, analytic on Re(z) >0and for all s ∈S. By (3.90), and our choice of the branch of the log function in (2.20), for all z∈C, Re(z) ≥z0>0 ˜ Vp(z,s)≤Ce−T 2Re(z)1 |z|ˆ Re(σ)=βe(σ−s)log(−z) B(σ)|dσ| 1−e2iπ(s−σ)≤Cz0e−T 2Re(z). (3.92) On the other hand, using the function ˜ Vhwe define, following the same rationale as in the definition of (2.22), in the Proof of Proposition (2.8) ˜ H(z,s)= ˜ Vh(z,s)eslog(−z) B(s) ˜ h(z,ζ)= ˜ H(z,s),ζ=e2iπ(s−β). For every zsuch that Re(z) >0, the function h(z, ·)is then analytic on C\R+and, by (3.91), ˜ h(z,ζ +i0) = ˜ h(z,ζ −i0),∀ζ∈R+. It follows that for all Re(z) >0, ˜ h(z, ·)is analytic on C\{0}. But since, by Proposition 2.4 and (3.92), we also have
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 49 |˜ h(z,ζ)|≤Ceslog(−z)=Ceclog zei(s−β)Arg(−z)=Ceclog z|ζ|Arg(−z) 2π=Ceclog z|ζ|1/2, by Liouville’s Theorem ˜ h(z) ≡0. Therefore ˜ H(z) =˜ Vh(z) =0and ˜ V=˜ Vp. By the inverse Laplace formula U(t, s)= 1 2iπ a+i∞ ˆ a−i∞ ˜ V(z,s)eztdz, and by (3.90)we have then U(t, s) =M(Λ(t, )(s) =0for all s ∈Sand 0 ≤t ≤T/2 from where the result follows. Proof of Theorem 1.2.All the properties of Λ, up to (1.25), have already been proved in Proposition 2.13, Corollary 2.14, Corollary 3.13, Corollary 3.15 and Proposition 3.18. Since W(2) =0, U(t, 3) =U(0, 3) for all t >0by (2.29), which is the conservation of the second moment of Λ(t). 4. Solution of the Cauchy problem for (1.1) This Section is devoted to the proof of the existence of solutions to the Cauchy problem for equation (1.1)for initial data f0∈L∞(0, ∞)or L1(0, ∞), and the proofs of Theorem 1.4 and Proposition 1.5. For all y>0we define, G(t, x;y)=y−1Λt y,x y,∀t>0,∀x>0.(4.1) By (3.71), G ∈C((0, ∞) ×(0, ∞); L1(0, ∞)), for y>0fixed it is a weak solution to (1.20)and lim t→0G(t, ·,y)=δy,in the weak sense of D(0,∞).(4.2) The function Galso satisfies the following important property, Proposition 4.1. There exists a positive constant CG>0such that, for all t >0, x >0, I(t, x)= ∞ ˆ 0|G(t, x;y)|dy < CG.(4.3) The proof of Proposition 4.1 has several auxiliary Lemmas and two different cases:
50 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 •If 0 <t <x, I(t, x)= t ˆ 0 (······) t/y > 1,x/y >1 dy y+ x ˆ t (······) t/y < 1,x/y >1 dy y+ ∞ ˆ x (······) t/y < 1,x/y <1 dy y.(4.4) •For 0 <x <t, I(t, x)= x ˆ 0 (······) t/y > 1,x/y >1 dy y+ t ˆ x (······) t/y > 1,x/y <1 dy y+ ∞ ˆ t (······) t/y < 1,x/y <1 dy y.(4.5) Lemma 4.2. There exists C>0such that, for all t >0and x >0, t ˆ 0Λt y,x y dy y≤C. Proof of Lemma 4.2.Since y∈(0, t), t/y > 1and by Proposition 3.1 and Proposition 3.2, Λt y,x y≤Cmax t y,x y−3 . Then, ∀x>0,∀t∈(0,x), t ˆ 0Λt y,x y dy y≤ t ˆ 0x y−3dy y=t3 3x3≤1/3. ∀t>0,∀x∈(0,t), t ˆ 0Λt y,x y dy y≤ t ˆ 0t y−3dy y=1 3 It remains now to estimate the two last integrals at the right hand side of (4.4), and the last one at the right hand side of (4.5). To this end we will be using a function δ(z), defined and continuous on z≥0such that, δis decreasing,δ(u)<1 for all u>0,δ(1) = 1 2,δ(u)=e1−u 2,∀u≥1 2.(4.6) 4.1. The domain 0 <t <x Consider first the domain where 0 <t <y<xwhere 0 <t y<1 <x y. In order to use the estimate on Λ, this domain is still subdivided.
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 51 Lemma 4.3. Define H2(z)=z(1 + δ(z)) and H1(z)=z(1 −δ(z)),∀z>0 These two functions are monotone increasing. Moreover ∀z>0,H 1(z)<z (4.7) ∀z>3/2,H −1 2(z)>1 (4.8) ∀z>0,H−1 2(z)<z (4.9) ∀x>0,∀t∈(0,2x/3),2x 3<tH −1 2x t.(4.10) Proof. Since the function H2is strictly increasing, its inverse H−1 2is well defined. The choice δ(1) =1/2makes H2(1) =3/2then H−1 2(3/2) =1. By monotonicity it follows that H−1 2(z) >H −1 2(3/2) =1for all z>3/2and this proves (4.8). Since H2(z) >zit follows that z>H −1 2(z)and this shows (4.10). Since δ(1) =1/2, we have 2 3(1 + δ(1)) = 1 and the function δ(z)is strictly decreasing because so is ρ(z). Therefore δ(z) <1/2for all z>1, and, for all t ∈(0, 2x/3) H22x 3t=2x 3t1+δ2x 3t<2x 3t(1 + δ(1)) = x t. Since H2is strictly increasing, so is H−1 2, 2x 3t≤H−1 2x tand this proves (4.10). Lemma 4.4. For all t >0, x >0such that t <x, x ˆ tΛt y,x y dy y≤C1+t+Φ 1+Ψ 1+˜ Φ2,(4.11) ∞ ˆ xΛt y,x y dy y≤C(1 + Φ3+Ψ 3),(4.12) where: Φ1(x, t)=t tH−1 2x t ˆ 2x 3 1 y x y−1 −1dy y,∀t∈(0,2x/3),(4.13) Ψ1(x, t)= x ˆ tH−1 2x t t y x y−1 −1+ 2t ydy y,∀t∈(0,2x/3),(4.14) ˜ Φ2(x, t)= x ˆ t t y x y−1 −1+ 2t ydy y,∀t∈(2x/3,x),(4.15)
52 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 Ψ3(x, t)= tH−1 1x t ˆ x t y x y−1 −1+ 2t ydy y,∀t∈(0,x),(4.16) Φ3(x, t)= 2x ˆ tH−1 1x t t y1−x y −1dy y,∀t∈(0,x).(4.17) Proof of Lemma 4.4.We show (4.11) first and start assuming t ∈(0, 2x/3). By (4.10), x ˆ tΛt y,x y;1 dy y= 2x 3 ˆ t (···)dy + tH−1 2x t ˆ 2x 3 (···)dy + x ˆ tH−1 2x t (···)dy. (4.18) In the first integral of the right hand side of (4.18), since y<2x/3, by Proposition 3.5 Λt y,x y≤C1x t−1−β 1t y6 +C2x t−6t y2 , and then, 2x 3 ˆ tΛt y,x y dy y≤C1t6 2x 3 ˆ t y−6dy +C2t2 2x 3 ˆ t y−2dy ≤Ct. (4.19) In the second integral of the right hand side of (4.18), simple computations yield, y∈2x 3,tH−1 2x t=⇒t yH2(y/t)<x y<3 2=⇒δy t<x y−1<1 2. Since x >3t/2we have y/t >1. On the other hand, x/t may take values arbitrarily large, and then H−1 2x tand y/t too. We deduce that δ(y/t) ∈(0, 1/2) and by Proposition 3.6, tH−1 2x t ˆ 2x 3Λt y,x y dy y≤CΦ1(x, t).(4.20) In the third integral of the right hand side of (4.18), since tH−1 2x t<y, it follows that tH−1 2x t<y, from where x t<H 2y t=y t1+δy t. Then x y<1 +δy tand, since x/y > 1also, 0<x y−1<δy t.(4.21) We notice now that since x/t >3/2and 3 2<x t=u(1 +δ(u)) ≤2u, we also have u =H−1 2(x/t) >3/4. Then y/t varies on the half line (3/4, ∞)and δ(y/t)varies on (0, δ(3/4)). We deduce from (4.21), using Corollary 3.13, that for some constant C>0,
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 53 Λt y,x y≤Ct y x y−1 −1+ 2t y .(4.22) It follows from (4.19), (4.20)and (4.22)that for 0 <t <2x/3, x ˆ tΛt y,x y dy y≤C(t+Φ 1(x, t)+Ψ 1(x, t)) .(4.23) Suppose now that t ∈(2x/3, x). We first deduce that since x/t <3/2and H−1 2is increasing, H−1 2(x/t) <H −1 2(3/2) =1and then tH−1 2(x/t) <t. Since y∈(t, x)it follows that y>tH −1 2(x/t)and therefore, H2(y/t)≡y t(1 + δ(y/t)) >x t=⇒1+δ(y/t)>x y⇐⇒ x y−1<δ(y/t). Then, for all 0 <t <y<x, we have x/y > 1 and, 0 <x y−1 <δ(y/t). By Corollary 3.14, and (4.15)we deduce, when t ∈(2x/3, x), x ˆ tΛt y,x y dy y≤˜ CΦ2(x, t),(4.24) and (4.11) follows from (4.23)and (4.24). We prove now (4.12). To this end we write, the left hand side as ∞ ˆ x (···)dy y= tH−1 1x t ˆ x (···)dy y+ 2x ˆ tH−1 1x t (···)dy y+ ∞ ˆ 2x (···)dy y(4.25) In the first term at the right hand side of (4.25)x <y<tH −1 1x t, then 0 <1 −x y<δy t from where, by Corollary 3.13 and (4.16) tH−1 1x t ˆ xΛt y,x y;1 dy y≤CΨ3(x, t),0<t<x. (4.26) In the second integral at the right hand side of (4.25), tH−1 1x t<y<2xand so δy t<1 −x y<1 2and by (4.17)and Proposition 3.6, 2x ˆ tH−1 1x tΛt y,x y dy y≤CΦ3(x, t)0<t<x. (4.27) In the last integral at the right hand side of (4.25), since y>2x, by Proposition 3.6,
54 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 ∞ ˆ 2xΛt y,x y dy y≤Ct ∞ ˆ 2x 1 |1−x/y| dy y2≤Ct ∞ ˆ 2x dz y2≤C. (4.28) The estimate (4.12) follows now by (4.26)–(4.28). 4.2. The domain 0 <x <t We estimate now the last integral at the right hand side of (4.5) Lemma 4.5. For all t >0and x ∈(0, t), ∀t>2x, ∞ ˆ tΛt y,x y dy y≤C(4.29) ∀t∈(x, 2x), ∞ ˆ tΛt y,x y dy y≤C(1 + Φ3+Ψ 4) (4.30) where Ψ4= tH−1 1x t ˆ t t y x y−1 −1+ 2t ydy y,∀t∈(x, 2x).(4.31) Proof of Lemma 4.5.If t >2xthen, x/y < 1/2and Proposition 3.6 gives (4.29). For t ∈(x, 2x), x t>1 2≡H1(1) and t <tH −1 1x tby the monotonicity of H1. On the other hand, H12x t=2x t1−δ2x t≥2x t(1 −δ(1)) = x t (where use has been made of 2x/t ≥1), and then, tH−1 1x t<2x. Therefore, ∞ ˆ t (···)dy = tH−1 1x t ˆ t (···)dy + 2x ˆ tH−1 1x t (···)dy + ∞ ˆ 2x (···)dy. (4.32) In the first term at the right hand side of (4.32)0 <1 −x y<δ y tbecause y∈ t, tH−1 1x t, tH−1 1x t ˆ tΛt y,x y dy y≤CΨ4(x, t),(4.33) by (4.31)and Corollary 3.14. In the second integral of the right hand side of (4.32)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 55 y∈tH−1 1x t,2x=⇒δy t<1−x y<1 2. By Proposition 3.6 and (4.17) 2x ˆ tH−1 1x tΛt y,x y dy y≤CΦ3(x, t).(4.34) In the third integral of the right hand side of (4.32)y>2xthen by Proposition 3.6, ∞ ˆ 2xΛt y,x y dy y≤C(4.35) and (4.30) follows from (4.32)–(4.35)for t ∈(x, 2x). 4.3. Estimates of the functions Φand Ψ In this sub Section some useful properties of the functions Φand Ψdefined in (4.13)–(4.17)are obtained. Lemma 4.6. There exists a constant C>0such that, Φ1+Ψ 1+˜ Φ2+Φ 3+Φ 4+Ψ 4≤C(4.36) Proof of Lemma 4.6.(i) Estimate of Φ1. By definition, for x >0and t ∈(0, 2x/3), Φ1(x, t)=Ct x t xH−1 2x t ˆ 2 3 |1−r|−1dr =−t xlog 1−t xH−1 2x t+log3 .(4.37) Then, for all ε >0, Φ1(x, t)is bounded for all (t, x)such that 0 <t <xand t xH−1 2x t∈ [0, 1 −ε]. Assume now that t xH−1 2x t→1, and denote u =H−1 2(x/t). Since, t xH−1 2x t=u H(u)=1 1+δ(u)(4.38) if t xH−1 2x t→1it follows that δ(u) →0. This implies that u →∞, and by elementary calculus, t xH−1 2x t=1 1+e1−u 2 =1−e1−u 2+Oe−2u,as u→∞ and t xH−1 2x t= u→∞ 1+Oe−u,u=H−1 2x t= u→∞ x t1+Oe−u.(4.39)
56 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 Using (4.38), (4.39)and the definition of δ, for ρ >0as small as desired and u →∞, t xH−1 2x t=1 1+e−x t(1 + Oe−(1−ρ)u)=1 1+e−x t1+Oe−(2−ρ)u and it follows that log 1−t xH−1 2x t= u→∞ −x t+Oe−x t. We deduce the existence of a constant C>0such that for all 0 <t <2x/3, Φ1(x, t)≤C. (4.40) (ii) Estimate of Ψ1. Since t ∈(0, 2x/3) and y>tH −1 2x tthen x/t <H 2(y/t) <2y/t. Using that y<x, also we deduce 0 <x y−1<1. Since 1/y > 1/x, Ψ1(x, t)≤t x ˆ tH−1 2x tx y−1−1+ 2t xdy y2=tx−1 1 ˆ t xH−1 2x t (1 −ρ)−1+ 2t xρ−1−2t xdρ By (4.10), 2H−1 2x t>4x 3t, then t xH−1 2x t>1 2and, Ψ1(x, t)≤tx−1 1 ˆ 1 2 (1 −ρ)−1+ 2t xρ−1−2t xdρ =C. (4.41) (iii) Estimate of ˜ Φ2. When t ∈(2x/3, x)and y∈(t, x), 0 <x y<1and then, by (4.15) ˜ Φ2(x, t)≤t x ˆ tx y−1−1+ 2t xdy y2=t x 1 ˆ t x (1 −r)−1+ 2t xr−1−2t xdr ≤t x 1 ˆ 2 3 (1 −r)−1+ 2t xr−1−2t xdr =2 −2t x−1≤2−4/3.(4.42) (iv) Estimate of Φ3. By definition, for 0 <t <x, Φ3(x, t)= t x 2 ˆ t xH−1 1x t (r−1)−1dr r=−t xlog t xH−1 1x t−1(4.43)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 57 because, if v=H−1 1x tthen x t=H1(v) =v(1 −δ(v)), and t xH−1 1x t=1 1−δ(v)>1. The same arguments as in the estimate of the right hand side of (4.37), show the existence of a constant C>0such that for all 0 <t <x, Φ3(x, t)≤C. (4.44) (v) Estimate of Ψ3. For all yin the domain of integration of Ψ3, y<tH −1 1x t, and then 2t y>2 H−1 1x t. Since y>xalso, we have 1−x y∈(0, 1) and we deduce from (4.16), Ψ3(x, t)≤t tH−1 1x t ˆ x1−x y−1+ 2 H−1 1x tdy y2=t x t xH−1 1x t ˆ 1 (r−1)−1+ 2 H−1 1x t r1+ 2 H−1 1x tdr We use now that, because δ(x/t) <1/2, z<H 1(2z)and so t xH−1 1x t<2, to obtain, Ψ3(x, t)≤t x 2 ˆ 1 (r−1)−1+ 2 H−1 1x t r1+ 2 H−1 1x tdr =t xH−1 1(x/t)2−1−2t H−1 1(x/t)≤C. (4.45) (vi) Estimate of Ψ4. By definition, x <t <y<tH −1 1x t<2x, for all yin the domain of integration. Therefore, as for Ψ3, we have 2t y>2 H−1 1x tand 1−x y∈(0, 1). Arguing as for Ψ3, we deduce from (4.31), for all t ∈(x, 2x), Ψ4(x, t)≤t tH−1 1x t ˆ t1−x y−1+ 2 H−1 1x tdy y2≤t 2 ˆ 1 (r−1)−1+ 2 H−1 1x t r1+ 2 H−1 1x tdr =tx−1H−1 1(x/t)2−1−2t H−1 1(x/t)≤C. (4.46) Lemma 4.6 follows from (4.40)–(4.46) Proof of Proposition 4.1.Proposition 4.1 follows from Lemmata 4.2–4.6 It is now possible to define the solution uof the Cauchy problem. 4.4. Proofs of Theorem 1.4 and Proposition 1.5 Theorem 4.7. (i) For any f0∈L1(0, ∞), ∞ ˆ 0 ∞ ˆ 0Λt y,x yf0(y) dy ydx < ∞,∀t>0.(4.47)
64 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 and since 1/2 < x/z < x/t1<3/2for all t ∈I, for a positive constant Cindependent on xand t1, ∂Λ ∂t t z,x z|f0(z)| z2≤C|f0(z)| z2.(4.69) A similar estimate holds for z∈Dc 1(t, x)with a similar argument. We are then left with the domain D(t, x), where, by (3.69)in Corollary 3.14 ∂Λ ∂t t z,x z|f0(z)| z2≤C(1 + 2(t/z)|log |(x/z)−1||)|f0(z)| |(x/z)−1|1−2(t/z)z2 ≤C(1 + |log |(x/z)−1||)|f0(z)| |(x/z)−1|1−2(t/z)z2 Since z∈D(t, x), E+(x/t) ≤z/t ≤E−(x/t)and |(x/z) −1| <1. Then, for t ∈(t1, t2), 1 E−(x/t1)≤t z≤1 E+(x/t2) =⇒|(x/z)−1|1−2t z≥|(x/z)−1|1−ρ(x,t1) ρ(x, t1)= 2 E−(x/t1)>0 ∂Λ ∂t t z,x z|f0(z)| z2≤C(1 + |log |(x/z)−1||)|f0(z)| |(x/z)−1|1−ρ(t1,x)z2. Notice that for z∈Dc, where (4.69)holds, x z−1≤1+x z≤1+ x 2t1 and, from (4.69), for z∈Dc(t, x)too, ∂Λ ∂t t z,x z|f0(z)| z2≤C(1 + |log |(x/z)−1||)|f0(z)| |(x/z)−1|1−ρ(t1,x)z2.(4.70) Therefore, when x/3 <t 0<3x 2the function H(x, z)may then be taken as follows, H(x, z)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Cz2|f0(z)| max(t4 1,x 4),∀z∈(0,2t1) (1 + |log |(x/z)−1||)|f0(z)| |(x/z)−1|1−ρ(t1,x)z2,∀z∈(2t1,3x) C|f0(z)| z2,∀z>3x, (4.71)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 65 where the constant Cmay depend on xand t1but not on tor z. We have then, shown that for all t0>0, and almost every x >0 there exists a neighborhood I=(t1, t2)such that, under the hypothesis... on f0, ∂Λ ∂t t z,x z|f0(z)| z2≤H(x, z) ∞ ˆ 0 H(x, z)dz < ∞. It follows from classical properties of Lebesgue’s integral, that for all t >0and a.e. x >0, ∂ ∂t ∞ ˆ 0 f0(z)Λ t z,x zdz z= ∞ ˆ 0 f0(z)∂tΛt z,x zdz z2,(4.72) and usatisfies (1.1)for all t >0and a.e. x >0. It is now possible, to obtain pointwise estimates of ∂tu(t, x), using essentially the same right hand side terms that in (4.57), (4.59), (4.71), except that tneeds not be replaced by t1or t2now. It easily follows, ∂u ∂t (t, x)≤Ct2x−4+t3x−3+ε||f0||1,a.e.x>3t C(t−2+t−3x)||f0||1,a.e.x∈(0,2t/3).(4.73) If x ∈(2t/3, 3t) denote, θ(t, x)=1−2 E−(x/t) and then, ˆ D(t,x) (1 + |log |(x/z)−1||)|f0(z)| |(x/z)−1|1−ρ(t1,x)z2≤sup z∈(2t,3x)|f0(z)|3x ˆ 2t C(1 + |log |(x/z)−1||) |(x/z)−1|1−ρ(t1,x)z2 and 3x ˆ 2t (1 + |log |(x/z)−1||) |(x/z)−1|θz2=1 x x/2t ˆ 1/3 (1 + |log |y−1||) |y−1|θ ≤1 (1 −θ)x2 31−θ +x 2t−1 1−θ+(2/3)1−θ(1 + (1 −θ)|log(2/3)|) (1 −θ)2x+
66 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 +|x/2t−1|1−θ(1 + (1 −θ)|log |x/2t−1||) (1 −θ)2x. Since x ∈(2t/3, 3t)it follows that x/2t ∈(−2/3, 1/2) and |x/2t −1| ≤2/3. Then, 3x ˆ 2t (1 + |log |(x/z)−1||) |(x/z)−1|θz2≤2 (1 −θ)x+(1 + |log(2/3)| (1 −θ)2x+(1 + |log |x/2t−1||) (1 −θ)2x ≤C(1 + |log |x/2t−1||) (1 −θ)2x and, ∂u ∂t (t, x)≤C(1 + |log |x/2t−1||) (1 −θ)2xsup z∈(2t,3x)|f0(z)|, a.e.x ∈(2t/3,3t).(4.74) We deduce from (4.73), (4.74)that L(u)satisfies (1.36). Proof of Proposition 1.5.When t >0is fixed and x →0we are in the region where 2x <tand we write, using the definition (1.30)of u, u(t, x)=I1+I2+I3,I 1= x ˆ 0 Λt y,x yf0(y)dy y, I2= t ˆ x Λt y,x yf0(y)dy y,I 3= ∞ ˆ t Λt y,x yf0(y)dy y. In the two first integrals of the right hand side t/y > 1, and then by (3.4), (3.10)and (3.12), for all δ>0as small as desired, Λt y,x y=t y−3 Q1x t+Q2t y,x t Q1x t=2c1B(1) W(0) +Ox t,x t→0 Q2t y,x t=c2t y−4 +b1t y+Oδt y−4x t 1−δ,x t→0,t y>1, =c2t y−4 +b1t y+t−4y4Ox t 1−δ,x t→0,t y>1. Therefore,
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 67 I1+I2= t ˆ 0 Λt y,x yf0(y)dy y=2c1B(1) W(0) t−3 t ˆ 0 f0(y)y2dy 1+Ox t+ +c2t−4 t ˆ 0 f0(y)y3dy + t ˆ 0 f0(y)b1t ydy y+t−4Oδx t 1−δt ˆ 0 f0(y)y3dy. Since 2x <t <yin I3, it follows that x/y < t/(2y), and by (3.27), (3.28) I3≤Cxt5 ∞ ˆ t |f0(y)|dy y7≤Cx1−δt5+δ ∞ ˆ t |f0(y)|dy y7. This concludes the proof of (1.37), (1.38), where b1is the function given in (3.15)and A1=−2 W(1)W(2)W(0),A 2=6˜ρ(2) W(0)W(3)W(1).(4.75) In order to prove (1.39)–(1.41), consider now 0 <x <xand write, u(t, x)−u(t, x)=I1+I2(4.76) I1= t ˆ 0Λt y,x y−Λt y,x yf0(y)dy y,(4.77) I2= ∞ ˆ tΛt y,x y−Λt y,x yf0(y)dy y(4.78) Three cases may now be considered, depending on whether x<x <t, t <x <xor x<t <x. Suppose first that x<x <t. Since t/y > 1in I1, by Proposition 3.4 and the mean value Theorem, ∃ξ=ξ(x, xy)∈x y,x y;|I1|≤|x−x| t ˆ 0 ∂Λ ∂x t y,ξ|f0(y)|dy y2 ≤|x−x|t−4 t ˆ 0|f0(y)|y2dy. (4.79) In I2, y>t >ρx >x from where, by Proposition 3.5, and again by the mean value Theorem, ∃ξ=ξ(x, xy)∈x y,x y;|I2|≤|x−x| ∞ ˆ t ∂Λ ∂x t y,ξ|f0(y)|dy y2
68 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 We use now Proposition 3.5 to obtain, |I2|≤C|x−x|t ∞ ˆ t ξ−1−σ∗ 0|f0(y)|y2dy ≤C|x−x|x−1−σ∗ 0t ∞ ˆ t|f0(y)|y−2+σ∗ 0dy, (4.80) where σ∗ 0∈(−2, −1) is defined in Proposition 2.1. Then, for x<x <t: |u(t, x)−u(t, x)|≤C|x−x|t−4 t ˆ 0|f0(y)|y2dy +C|x−x|x−1−σ∗ 0t ∞ ˆ t|f0(y)|y−2+σ∗ 0dy, (4.81) and this shows (1.39). Suppose now that x >x >t. By a similar argument as before, using now Proposition 3.4, |I1|≤C|x−x|t2 x4 t ˆ 0|f0(y)|dy. (4.82) The term I2must be decomposed in three integrals. Two of them are estimated as in the previous case using Proposition 3.5 x ˆ tΛt y,x y−Λt y,x yf0(y)dy y≤|x−x| x ˆ t ∂Λ ∂x t y,ξ(y)|f0(y)|dy y2 ≤C|x−x|t−1x−1 x ˆ t|f0(y)|dy (4.83) and ∞ ˆ xΛt y,x y−Λt y,x yf0(y)dy y≤|x−x| ∞ ˆ x ∂Λ ∂x t y,ξ(y)|f0(y)|dy y2 ≤C|x−x|tx−4 ∞ ˆ x|f0(y)|dy. (4.84) The last integral is, J(t, x, x)= x ˆ xΛt y,x y−Λt y,x yf0(y)dy y.
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 69 The integration interval goes from xto x, and then t/y < 1and x/y < 1 <x/y. We must then use the Proposition 3.10: Λt y,x y−Λt y,x y≤2|x−x|1−α x1−α|log(x/y)|1−αΛt y,x y +C|x−x|r−α xr−α|log(x/y)|(1+α)(r−α)(4.85) r=t 2x,α=(M−2)t 2Mx ∈(0,r).M>3=⇒(1 + α)(r−α)≤3 2M<1 2(4.86) from where, |J(t, x, x)|≤2|x−x|1−α x1−α x ˆ xΛt y,x y|f0(y)|dy y|log(x/y)|1−α+ +C|x−x|r−α xr−α x ˆ x |f0(y)|dy y|log(x/y)|(1+α)(r−α)(4.87) Since (1 +α)(r−α) ∈(0, 1), if f0∈L∞ loc(0, ∞), ∞ ˆ 0 |f0(y)|1(x,x)(y)dy y|log(x/y)|(1+α)(r−α)≤||f0||L∞(x,x) x/x ˆ 1 dz z|log(z)|(1+α)(r−α) =||f0||L∞(x,x) 1−(1 + α)(r−α)(log(x/x))1−(1+α)(r−α)≤||f0||L∞(x,x)(log(x/x))1−(1+α)(r−α) ≤||f0||L∞(x,x)(1 + log(x/x)).(4.88) By similar arguments and (4.49)in Theorem 4.7 % % % %|f0(y)1(x,x)|dy y|log(x/y)|1−α% % % %1≤ ∞ ˆ x |f0(y)|dy y|log(x/y)|1−α≤||f0||L∞(x,∞) 3x ˆ x dy y|log(x/y)|1−α+ +1 3|log 3|1−αx ∞ ˆ 3x|f0(y)|dy ≤||f0||L∞(x,3x) 2x t(log 3)t/2x+1 3|log 3|1−αx ∞ ˆ 3x|f0(y)|dy (4.89) and then, x ˆ xΛt y,x y|f0(y)|dy y|log(x/y)|1−α
70 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 ≤C||f0||L∞(x,3x) x t(log 3)t/2x+1 |log 3|1−αx ∞ ˆ 3x|f0(y)|dy.(4.90) It follows from, (4.83), (4.84), (4.87))–(4.90) |I2|≤C|x−x|t−1x−1 x ˆ t|f0(y)|dy +C|x−x|tx−4 ∞ ˆ x|f0(y)|dy+ +2|x−x|1−α x1−α||f0||L∞(x,3x) x t(log 3)t/2x+1 |log 3|1−αx ∞ ˆ 3x|f0(y)|dy+ +C|x−x|r−α xr−α||f0||L∞(x,x)(1 + log(x/x)) (4.91) Then, since 1 −α∈(0, 1/2) and t/2x <1/2 |u(t, x)−u(t, x)|≤C|x−x|t2 x4 t ˆ 0|f0(y)|dy +1 tx x ˆ t|f0(y)|dy +t x4 ∞ ˆ x|f0(y)|dy+ +2|x−x|1−α x1−α||f0||L∞(x,3x) x t+1 x ∞ ˆ 3x|f0(y)|dy+ +C|x−x|r−α xr−α||f0||L∞(x,x)(1 + log(x/x)),∀x>x >t>0,(4.92) and this shows (1.41). Assume now that x<t <x. Then, in the first term I1we use that for all τ>1and z>0, ∂Λ ∂x(τ,z)=−1 4π2ˆ Re(s)=c sx−s−1U(τ,s)ds (cf. (2.27)and (3.20)), and by Proposition 2.10, for all c ∈(0, 2) there exists a numerical constant C=C(c) >0such that, ∂Λ ∂x(τ,z)≤Cx−1−cˆ R |s|(1 + |s|)−2τds ≤Cx−1−c(1 + τ2)−1. We have then, using the same notation ξ∈(x/y, x/y), |I1|≤C|x−x| t ˆ 0 ξ−1−c|f0(y)|dy y(1 + (t/y)2)≤C|x−x| x1+ct1−c t ˆ 0|f0(y)|dy. (4.93)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 71 On the other hand, we split the I2term as I2= x ˆ t···dy + ∞ ˆ x···dy =I2,1+I2,2 The term I2,2is estimated exactly as in (4.84). In the term I2,1estimates (4.85)and (4.86)are used again to obtain, |I2,1|≤ 2|x−x|1−α x1−α|log(x/t)|1−α x ˆ tΛt y,x y|f0(y)|dy y +C|x−x|r−α txr−αlog(x/t)|(1+α)(r−α) x ˆ t|f0(y)|dy ≤C||f0||1|x−x|1−α x1−α|log(x/t)|1−α+|x−x|r−α txr−αlog(x/t)|(1+α)(r−α).(4.94) Then, if x<t <x, for all c ∈(0, 2) there exists a constant Csuch that, |u(t, x)−u(t, x)|≤C|x−x|1 x1+ct1−c t ˆ 0|f0(y)|dy +t x4 ∞ ˆ x|f0(y)|dy+ +C||f0||1|x−x|1−α x1−α|log(x/t)|1−α+|x−x|r−α txr−αlog(x/t)|(1+α)(r−α).(4.95) And this proves (1.40). On the other hand, by (2.37)and (4.48), for all t >0and sfixed, M(u(t))(s)= ∞ ˆ 0 Ut y,s f0(y)ys−1. Since B(3) =0(cf. Proposition (2.3)), properties (1.42)and (1.43) follow from (2.31). 5. Appendix 5.1. The proof of Proposition 2.10 Proof. Based on the expression (2.27)of U(t) U(t, s)=B(s) 2iπ ˆ Re(σ)=β t−(σ−s)Γ(σ−s) B(σ)dσ, β ∈(0,2)
72 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 the proof closely follows that of Proposition 8.1 in [13] (similar to (5.1) in [13]). As in (8.34) of [13], this may be written, U(t, s)=B(s) 2iπ ˆ Re(Y)=β−Re(s) t−YΓ(Y) B(s+Y)dY =ˆ Re(σ)=β eψ(s,σ,t)A(Y)dY (5.1) where Ψ(s, Y, t)= ˆ Re(ρ)=β log (−W(ρ)) Θ(ρ−s, Y )dρ −Ylog t−Y+Y−1 2log Y, (5.2) with Θ defined in (2.19), and A(Y)= Γ(Y) 2iπe−YYY−1/2.(5.3) The function Adefined in (5.3)is the same as in (8.5) of [13], up to the constant factor −i(2π)−1/2. The function Ψ defined in (5.2)is similar to (8.4) in [13], the only difference lies in the function Winstead of Φ. The proof of the estimates (2.32), (2.33)of Proposition 2.10 follows then the same arguments as in [13]with only minor differences. For sin bounded sets, contour deformation and method of residues in the integrals (5.1), (5.2). For |s|large, these arguments are combined with the stationary phase Theorem applied to Ψ(s, Y, t)as a function of Y, where sand tare fixed. The variable Yis scaled as Y=2Zlog |s|, according to the behavior of W(s)as Im(s) →∞, for Re(s)in a fixed bounded interval and the result follows from the following. If we define, ˜ F(s, ζ)= ˆ Re(ρ)=β log (−W(ρ)) Θ(ρ−s, ζ)dρ (5.4) F(s, Z)= ˆ Re(ρ)=β log (−W(ρ)) Θ(ρ−s, 2Zlog |s|)dρ, =˜ F(s, 2Zlog |s|) (5.5) Estimates (2.34)and (2.35) follow now using (2.4)and (2.5). Let us define, TL=S0,2∪{s∈C:Re(s)≤L|s|≥2L}(5.6) where S0,2, defined as S0,2={s ∈C; Re(s) ∈(0, 2)}, is the region of analyticity of U(t).
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 73 Lemma 5.1. For any constant C>0, there exists a constant L >0and s0∈C, both depending on C, such that, for all s ∈TL∩Bs0(0)cthe function Fmay be extended analytically for Z∈D(s, C) ∩B|log |s|| 8(0) where D(s, C)=&s∈C,Re(s)<0,|Re(s)|≤C|Im(s)+|log |s|| 8|' There also exists a constant C>0, that depends on C, such that, for all Z∈D1(s, C) ∩ B|log |s|| 8(0) and s ∈TL∩Bs0(0)c, |F(s, Z)+Zlog(−W(s)) log |s|| ≤ CZ2+O1 log |s|.(5.7) Proof. The Proof of (5.1) closely follows that of Lemma 14.1 in [13]. The function Fis extended as analytical function on D(s, C) ∩B|log |s|| 8(0) by a modification of the representation formula (5.5)using contour deformation. The integral in the new integration contour Cis then written as ˆ C log (−W(ρ)) Θ(ρ−s, 2Zlog |s|)dρ =log(−W(s)) ˆ C Θ(ρ−s, 2Zlog |s|)dρ+ +ˆ C log W(ρ) W(s)Θ(ρ−s, 2Zlog |s|)dρ. The first integral may be explicitly calculated. The second is estimated using the cut off properties of the function Θand elementary calculus arguments completely similar to those of Lemma 14.1 in [13]. Due to the slow decay of the function U(t, s)as |s| →∞, the following is also needed Lemma 5.2. There exists a constant C>0such that, for all s ∈TL∩Bs0(0)c, and ζ such that Z=ζ/|s|∈D1(s, C) ∩B|log |s|| 8(0), ∂˜ F ∂s (s, ζ)≤C|ζ|2 |s|2log |s|+Ce−a|s|(5.8) Proof. By (5.5) ∂˜ F ∂s (s, ζ)= ˆ Re(r)=β−Re(s) ∂ ∂s (log (−W(r+s))) Θ(r, ζ)dr =−ˆ Re(r)=β−Re(s) W(r+s) W(r+s)Θ(r, ζ)dr =ˆ Re(ρ)=β W(ρ) W(ρ)Θ(ρ−s, ζ)dρ
80 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 On the other hand, by Proposition (2.5), |B(s)|≥C|log |s||,∀s, |s|≥R Then, for Re(s)on any compact subset of (0, 2), the function |B(s)|is uniformly bounded from below by a positive constant. It follows, for |ζ| ≥δ0/t2, and tsmall, ∂˜ F ∂s (σ/ρ(t),ζ)e˜ F(σ/ρ(t),ζ)Γ(ζ)t−ζ= ∂˜ F ∂s (σ/ρ(t),ζ) Bσ ρ(t) Bσ ρ(t)+ζΓ(ζ)t−ζ ≤C(1 + |ζ|)ρ(t)2t−4 |σ|2log |σ/ρ(t)|+e−a|σ/ρ(t)||log |σ/ρ(t)||e−|π||ζ| 2e−(β1−α1)logt ≤C(1 + |ζ|)ρ(t)2t−4 ε2 0(t−1+logε0)+e−a|σ/ρ(t)|(log M+t−1)e−|π||ζ| 2e(β1−α1)|log t| ≤C(1 + |ζ|)ρ(t)2t−4+e−aε0/ρ(t)(log M+t−1)e−|π| 4t2e(β1−α1)|log t|e−|π||ζ| 4, and |J3|≤Cρ(t)2t−4+e−aε0/ρ(t)(log M+t−1)e−|π| 4t2ˆ Re(ζ)=β1−α1 Im(ζ)≥δ0 t2 (1 + |ζ|)e−|π||ζ| 4dζ ≤Cρ(t)2t−4+e−aε0/ρ(t)(log M+t−1)e−|π| 4t2 Therefore, uniformly for |σ| ∈(ε0, M), log M∈(0, t−θ), lim t→0ρ(t)−1|J3|=0. Proceeding similarly with ∂U/∂t, since for β∈(0, 2) such that β−1 <c <β, ∂ ∂tU(t, s)= 1 2iπ ˆ Re(ζ)=β−Re(s) e˜ F(s,ζ)t−ζ−1Γ(ζ+1)dζ (5.31) it follows, ∂ ∂s ∂U ∂t t, σ ρ(t)=1 2iπ ˆ Re(ζ)=β1−α1 ∂˜ F ∂s σ ρ(t),ζe˜ Fσ ρ(t),ζΓ(ζ+1)t−ζ−1dζ, from where (5.27)is deduced with the same arguments used to obtain (5.25).
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 81 Lemma 5.5. Ht, σ ρ(t)=−tρ(t) 2σexp −2tlog bσ ρ(t). H1t, σ ρ(t)=∂H ∂t t, σ ρ(t) Proof. The integral in (5.26)can be computed adding the residues of the integrand at the poles ζ=−nof the Gamma function, Ht, σ ρ(t)=ρ(t) 2σlog |bσ/ρ(t) 2| ∞ n=0 (−1)ntn n!nexp nlog 2log bσ ρ(t) =−tρ(t) 2σexp −2tlog bσ ρ(t). On the other hand, H1t, σ ρ(t)=ρ(t) 2σlog |bσ/ρ(t) 2| ∞ n=0 (−1)ntn n!(n+1)exp(n+1)log2log bσ ρ(t) = exp −2tlog bσ ρ(t)ρ(t) 2σ−2log bσ ρ(t) texp −2tlog bσ ρ(t)ρ(t) 2σ =∂H ∂t t, σ ρ(t). Proposition 5.6. M−1(H(t))(X)=−2t πΓ(−2t)sin(πt)|X|2tsign(X). Proof. If we call X=ρ(t)Y, M−1(H(t))(X)= 1 2iπ ˆ Re(s)=α1 H(t, s)e−sρ(t)Yds =1 2iπρ(t)ˆ Re(σ)=α1ρ(t) Ht, σ ρ(t)e−σY dσ =t 4iπ ˆ Re(σ)=α1ρ(t) σ−1exp −2tlog bσ ρ(t)e−σY dσ we deform the integration contour to Re(σ) =0, and change variables bv →v,
82 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 t 4iπ ˆ Re(σ)=0 σ−1e−2tlogbv ρ(t)e−ivY dσ =t 4πˆ R v−1e−2tlogv ρ(t)e−iv Y bdv Then, after the change of variables v=ρ(t)w, dv =ρ(t)dw, t 4πˆ R v−1exp −2tlog v ρ(t)e−iv Y bdv =t 4πˆ R v−1exp (−2tlog |w|)e−iw ρ(t)Y bdw =−2t πΓ(−2t)sin(πt)|X|2tsign(X). Proof of Proposition 3.11.We use (3.50)to write the left hand side of (3.65)as, t−1|X|1−2t˜ Λ(t, X)=t−1|X|1−2tX−1X˜ Λ(t, X) =1 2iπ t−1|X|1−2tX−1ˆ Rr(s)=α1 ∂U ∂s (t, s)e−sXds. For X=ρ(t)Y, ˆ Rr(s)=α1 ∂U ∂s (t, s)e−sXds =1 2iπρ(t)ˆ Re(σ)=α1ρ(t) ∂U ∂s t, σ ρ(t)e−σY dσ (5.32) ˆ Re(σ)=α1ρ(t) ∂U ∂s t, σ ρ(t)e−σY dσ =I1+I2+I3(5.33) Ik=1 2iπ ˆ Re(σ)=α1ρ(t) σ∈Dk ∂U ∂s t, σ ρ(t)e−σY dσ (5.34) D1=Bε0(0),D 2=BM(t)(0) \Bε0(0),D 3=BM(t)(0)c(5.35) where log M(t) =t−3/2. On D1and D3we use (2.33)of Proposition 2.10, ∂U ∂s t, σ ρ(t)≤CTte−2tlog(|bσ/ρ(t)|)1+ σ ρ(t)−1 ≤Cte−2tlog |bv|e2tlog(ρ(t)) 1+ σ ρ(t)−1 ≤Ctρ(t)|σ|−2t−1, from where, |I1|≤Ctρ(t)ε0,|I3|≤Cρ(t)M(t)−2t.(5.36)
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 83 On D2, I2=I2,1+I2,2 I2,1=1 2iπ ˆ Re(σ)=α1ρ(t) σ∈D2 ∂U ∂s t, σ ρ(t)−Ht, σ ρ(t)e−σY dσ I2,2=1 2iπ ˆ Re(σ)=α1ρ(t) σ∈D2 Ht, σ ρ(t)e−σY dσ The first integral is estimated as |I2,1|≤ 1 2iπ ˆ Re(σ)=α1ρ(t) σ∈D2 ∂U ∂s t, σ ρ(t)−Ht, σ ρ(t)|dσ|. and by Lemma 5.4: lim t→0ρ(t)−1|I2,1|=0.(5.37) We write the second as I2,2−1 2iπ ˆ Re(σ)=α1ρ(t) Ht, σ ρ(t)e−σY dσ≤C ˆ Re(σ)=α1ρ(t) σ∈D1 Ht, σ ρ(t)e−σY dσ + +C ˆ Re(σ)=α1ρ(t) σ∈D3 Ht, σ ρ(t)e−σY dσ (5.38) and the expression of H(t)gives, by calculations similar to those giving (5.36), ˆ Re(σ)=α1ρ(t) σ∈D1 Ht, σ ρ(t)e−σY dσ ≤Ctρ(t)ε0(5.39) ˆ Re(σ)=α1ρ(t) σ∈D3 Ht, σ ρ(t)e−σY dσ ≤Ctρ(t)M(t)−2t(5.40) It follows from (5.33)and (5.36)–(5.40)that for all ε0>there exists τsmall enough such that, for all t ∈(0, τ)and all Y≥0,
84 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 t−1ρ(t)−1|I1|+|I3|+|I2,1|+I2,2−ρ(t)(M−1(H(t))(ρ(t)Y)≤ ≤Cε0+t−1M(t)−2t and then, uniformly on Y∈R, lim t→0t−1ρ(t)−1|I1|+|I3|+|I2,1|+I2,2−ρ(t)(M−1(H(t))(ρ(t)Y)=0.(5.41) Therefore, since for X=ρ(t)Y ˆ Re(s)=α1 ∂U ∂s (t, s)e−sXds =ρ(t)−1(I1+I2+I3) =ρ(t)−1(I1+I3+I2,1+I2,2−ρ(t)(M−1(H(t))(X))+(M−1(H(t))(X) and, t−1X−1|X|1−2tˆ Re(s)=α1 ∂U ∂s (t, s)e−sXds =t−1X−1|X|1−2tρ(t)−1(I1+I3+I2,1+ +I2,2−ρ(t)M−1(H(t)(X)+t−1X−1|X|1−2tM−1(H(t))(X) and by (5.41)we deduce, lim t→0t−1X−1|X|1−2tˆ Re(s)=α1 ∂U ∂s (t, s)e−sρ(t)Yds = lim t→0|X|1−2tM−1(H(t))(X) tX =1 uniformly for Xin bounded subsets of R. Property (3.60) follows for tsufficiently small, and then for t ∈(0, 1). The same arguments give (3.63)and then (3.61). 5.3. Linearization: the equation (1.15) When R(p, p1, p2) −R(p1, p, p2) −R(p2, p1, p)is written in terms of the function Ω defined in (1.10)and only linear terms in Ωare kept, the result is n0(1 + n0)∂Ω(t) ∂t =nc(t)LI3(Ω(t)) (5.42) LI3(Ω(t)) = ∞ ˆ 0 (U(k,k)Ω(t, k)−V(k,k)Ω(t, k)) k2dk,(5.43) 1 8nca2m−2U(k,k)=(mθ(k−k) kk ×n0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))] + (k↔k))
M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 85 −m kkn0(ω(k)+ω(k))[1 + n0(ω(k))][1 + n0(ω(k))],(5.44) 1 8nca2m−2V(k,k)=mθ(k−k) kkn0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))] +mθ(k−k) kkn0(ω(k))[1 + n0(ω(k))][1 + n0(ω(k)−ω(k))] (5.45) where k=|p|and k=|p|. The functions U(k, k)and V(k, k)have a non integrable singularity along the diagonal k=k. However, these singularities cancel each other when the two terms are combined as in (5.43)as far as it is assumed that, for all t >0, Ω(t) ∈Cα(0, ∞)for some α>0. But the integrand (U(k, k)Ω(t, k)−V(k,k)Ω(t, k)) can not be split as for the linearized Boltzmann equations for classical particles ([7]). However, an explicit calculation shows that, for all k>0, LI3(ω)(k)= ∞ ˆ 0U(k,k)k2−V(k,k)k2k2dk= 0 (5.46) from where we deduce, for all k>0, ∞ ˆ 0U(k,k)k2 k2Ω(t, k)−V(k,k)Ω(t, k)k2dk=Ω(t, k) kLI3(ω)(k)=0. We may then write, LI3(Ω(t)) = nc(t) ∞ ˆ 0 (U(k,k)Ω(t, k)−V(k,k)Ω(t, k)) k2dk =nc(t) ∞ ˆ 0 U(k,k)Ω(t, k) k2−Ω(t, k) k2k2k2dk and the linearized equation reads, n0(1 + n0)∂Ω(t) ∂t =nc(t) ∞ ˆ 0 U(k,k)Ω(t, k) k2−Ω(t, k) k2k2k2dk.(5.47) Use of the change of variables (1.10)-(1.11)in (5.47) yields equation (1.12)for the function u.
86 M. Escobedo / Journal of Functional Analysis 282 (2022) 109390 5.4. From (1.1)to (1.20) If uis a regular function, the right hand side of the equation (1.1)may be written, ∞ ˆ 0 (u(y)−u(x))K(x, y)dy)= ∞ ˆ 0 y ˆ x ∂u ∂z(z)dzK(x, y)dy =− x ˆ 0 ∂u ∂z(z) z ˆ 0 K(x, y)dydz + ∞ ˆ x ∂u ∂z(z) ∞ ˆ z K(x, y)dydz = ∞ ˆ 0 ∂u ∂z(z)Hx zdz z(5.48) Hx z=1z>x ∞ ˆ z K(x, y)dy −10<z<x z ˆ 0 K(x, y)dy (5.49) where an explicit integration of the two integrals in the right hand side of (5.49)gives (1.21), and then, the right hand side of equation (1.20). Acknowledgments The research of the author is supported by grants PID2020-112617GB-C21 of MINECO and IT1247-19 of the Basque Government. The hospitality of IAM of the University of Bonn, and its support through SFB 1060 are gratefully acknowledged. The author thanks Pr. M. Valle at the Universidad del País Vasco (UPV/EHU) for enlightening discussions. The author is also grateful to the referees for their careful reading of the manuscript, their comments and suggestions. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964, xiv+1046. [2] L. Arkeryd, R. Esposito, M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Commun. Math. Phys. 111 (1987) 393–407. [3] L. Arkeryd, A. Nouri, Bose condensates in interaction with excitations: a kinetic model, Commun. Math. Phys. 310 (2012) 765–788. [4] J. Banasiak, W. Lamb, Ph. Laurençot, Analytic Methods for Coagulation Fragmentation Models, Vol I. Chapman and Hall/CRC, New York; Vol II. CRC Press, Boca Raton, 2019. [5] M.J. Bijlsma, E. Zaremba, H.T.C. Stoof, Condensate growth in trapped Bose gases, Phys. Rev. A 62 (2000) 063609. [6] C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein, Bogoliubov theory in the Gross–Pitaevskii limit, Acta Math. 222 (2) (2019) 219–335.
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